{"segments": [{"id": 1, "seek": 3609, "start": 7.39, "end": 36.09, "text": " So last time, we discussed how to find the probabilities underneath the normal curve for three cases. If the point lies in the lower tail, as this one, or upper tail, or between. So we can do the computations for this kind of probabilities. And I think we gave two examples.", "tokens": [407, 1036, 565, 11, 321, 7152, 577, 281, 915, 264, 33783, 7223, 264, 2710, 7605, 337, 1045, 3331, 13, 759, 264, 935, 9134, 294, 264, 3126, 6838, 11, 382, 341, 472, 11, 420, 6597, 6838, 11, 420, 1296, 13, 407, 321, 393, 360, 264, 2807, 763, 337, 341, 733, 295, 33783, 13, 400, 286, 519, 321, 2729, 732, 5110, 13], "avg_logprob": -0.1269211046031264, "compression_ratio": 1.5536723163841808, "no_speech_prob": 0.0, "words": [{"start": 7.39, "end": 7.79, "word": " So", "probability": 0.7919921875}, {"start": 7.79, "end": 7.99, "word": " last", "probability": 0.76611328125}, {"start": 7.99, "end": 8.19, "word": " time,", "probability": 0.8916015625}, {"start": 8.27, "end": 8.37, "word": " we", "probability": 0.9638671875}, {"start": 8.37, "end": 8.87, "word": " discussed", "probability": 0.88134765625}, {"start": 8.87, "end": 9.11, "word": " how", "probability": 0.9453125}, {"start": 9.11, "end": 9.27, "word": " to", "probability": 0.970703125}, {"start": 9.27, "end": 9.55, "word": " find", "probability": 0.888671875}, {"start": 9.55, "end": 9.89, "word": " the", "probability": 0.912109375}, {"start": 9.89, "end": 10.59, "word": " probabilities", "probability": 0.8779296875}, {"start": 10.59, "end": 11.25, "word": " underneath", "probability": 0.94677734375}, {"start": 11.25, "end": 12.29, "word": " the", "probability": 0.91259765625}, {"start": 12.29, "end": 12.69, "word": " normal", "probability": 0.875}, {"start": 12.69, "end": 13.11, "word": " curve", "probability": 0.9052734375}, {"start": 13.11, "end": 14.57, "word": " for", "probability": 0.880859375}, {"start": 14.57, "end": 14.83, "word": " three", "probability": 0.8818359375}, {"start": 14.83, "end": 15.25, "word": " cases.", "probability": 0.9443359375}, {"start": 16.01, "end": 16.81, "word": " If", "probability": 0.96435546875}, {"start": 16.81, "end": 16.95, "word": " the", "probability": 0.91845703125}, {"start": 16.95, "end": 17.21, "word": " point", "probability": 0.97216796875}, {"start": 17.21, "end": 17.67, "word": " lies", "probability": 0.93359375}, {"start": 17.67, "end": 19.11, "word": " in", "probability": 0.92041015625}, {"start": 19.11, "end": 19.43, "word": " the", "probability": 0.91455078125}, {"start": 19.43, "end": 20.05, "word": " lower", "probability": 0.87060546875}, {"start": 20.05, "end": 20.45, "word": " tail,", "probability": 0.826171875}, {"start": 21.69, "end": 22.37, "word": " as", "probability": 0.869140625}, {"start": 22.37, "end": 22.59, "word": " this", "probability": 0.94921875}, {"start": 22.59, "end": 22.87, "word": " one,", "probability": 0.92333984375}, {"start": 23.69, "end": 24.15, "word": " or", "probability": 0.955078125}, {"start": 24.15, "end": 24.55, "word": " upper", "probability": 0.787109375}, {"start": 24.55, "end": 24.99, "word": " tail,", "probability": 0.8525390625}, {"start": 26.09, "end": 26.47, "word": " or", "probability": 0.96630859375}, {"start": 26.47, "end": 26.83, "word": " between.", "probability": 0.89990234375}, {"start": 27.55, "end": 27.83, "word": " So", "probability": 0.94970703125}, {"start": 27.83, "end": 27.95, "word": " we", "probability": 0.9130859375}, {"start": 27.95, "end": 28.15, "word": " can", "probability": 0.94677734375}, {"start": 28.15, "end": 28.51, "word": " do", "probability": 0.96142578125}, {"start": 28.51, "end": 29.03, "word": " the", "probability": 0.91552734375}, {"start": 29.03, "end": 29.81, "word": " computations", "probability": 0.97705078125}, {"start": 29.81, "end": 30.45, "word": " for", "probability": 0.94970703125}, {"start": 30.45, "end": 30.79, "word": " this", "probability": 0.90966796875}, {"start": 30.79, "end": 31.23, "word": " kind", "probability": 0.900390625}, {"start": 31.23, "end": 32.29, "word": " of", "probability": 0.97119140625}, {"start": 32.29, "end": 33.05, "word": " probabilities.", "probability": 0.8662109375}, {"start": 33.25, "end": 33.39, "word": " And", "probability": 0.93603515625}, {"start": 33.39, "end": 33.79, "word": " I", "probability": 0.7509765625}, {"start": 33.79, "end": 34.07, "word": " think", "probability": 0.9130859375}, {"start": 34.07, "end": 34.25, "word": " we", "probability": 0.94970703125}, {"start": 34.25, "end": 34.53, "word": " gave", "probability": 0.806640625}, {"start": 34.53, "end": 35.69, "word": " two", "probability": 0.92919921875}, {"start": 35.69, "end": 36.09, "word": " examples.", "probability": 0.86865234375}], "temperature": 1.0}, {"id": 2, "seek": 6120, "start": 36.62, "end": 61.2, "text": " For example, if we are looking for probability of Z is smaller than 0.1, and Z, as we mentioned last time, is the standardized normal distribution that has mean of 0 and sigma is 1. In this case, just go to the table you have. Now we are looking for 0.12, for example. 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This value is the probability of Z smaller than 0.5. But you have to keep in mind that we have to transform first from normal distribution to standardized normal distribution. For this specific example, if we are looking for mean of 8 and standard deviation of 5, that's for the normal distribution.", "tokens": [6974, 568, 11, 321, 483, 341, 1874, 13, 639, 2158, 307, 264, 8482, 295, 1176, 4356, 813, 1958, 13, 20, 13, 583, 291, 362, 281, 1066, 294, 1575, 300, 321, 362, 281, 4088, 700, 490, 2710, 7316, 281, 31677, 2710, 7316, 13, 1171, 341, 2685, 1365, 11, 498, 321, 366, 1237, 337, 914, 295, 1649, 293, 3832, 25163, 295, 1025, 11, 300, 311, 337, 264, 2710, 7316, 13], "avg_logprob": -0.19146286318267602, "compression_ratio": 1.5893719806763285, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 62.52, "end": 62.92, "word": " Under", "probability": 0.50390625}, {"start": 62.92, "end": 63.3, "word": " 2,", "probability": 0.5791015625}, {"start": 63.48, "end": 63.7, "word": " we", "probability": 0.94970703125}, {"start": 63.7, "end": 63.92, "word": " get", "probability": 0.9287109375}, {"start": 63.92, "end": 64.14, "word": " this", "probability": 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So 8.6 minus 8 divided by 5 gives 0.12. In this case, we can use the standardized normal table. Now, for the other case, we are looking for the probability of x greater than 8.6. So we are looking for the upper tail probability. 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274.33, "word": " the", "probability": 0.90966796875}, {"start": 274.33, "end": 275.15, "word": " probability", "probability": 0.94921875}, {"start": 275.15, "end": 275.31, "word": " of", "probability": 0.89892578125}, {"start": 275.31, "end": 275.43, "word": " z", "probability": 0.93603515625}, {"start": 275.43, "end": 275.81, "word": " smaller", "probability": 0.6630859375}, {"start": 275.81, "end": 276.11, "word": " than", "probability": 0.93701171875}, {"start": 276.11, "end": 276.45, "word": " 0", "probability": 0.82421875}, {"start": 276.45, "end": 277.81, "word": " is", "probability": 0.6767578125}, {"start": 277.81, "end": 277.95, "word": " one", "probability": 0.55712890625}, {"start": 277.95, "end": 278.17, "word": "-half.", "probability": 0.734130859375}], "temperature": 1.0}, {"id": 11, "seek": 30702, "start": 278.86, "end": 307.02, "text": " Because the total area underneath the normal curve is 1, and 0 divided the curve into two equally parts. So the area to the right of 0 is the same as the area to the left of 0. So in this case, minus 0.5. So this is your answer. So the probability of Z between 8 and 8.6 is around 0478. I think we stopped last time at this point.", "tokens": [1436, 264, 3217, 1859, 7223, 264, 2710, 7605, 307, 502, 11, 293, 1958, 6666, 264, 7605, 666, 732, 12309, 3166, 13, 407, 264, 1859, 281, 264, 558, 295, 1958, 307, 264, 912, 382, 264, 1859, 281, 264, 1411, 295, 1958, 13, 407, 294, 341, 1389, 11, 3175, 1958, 13, 20, 13, 407, 341, 307, 428, 1867, 13, 407, 264, 8482, 295, 1176, 1296, 1649, 293, 1649, 13, 21, 307, 926, 1958, 14060, 23, 13, 286, 519, 321, 5936, 1036, 565, 412, 341, 935, 13], "avg_logprob": -0.15082721219343298, "compression_ratio": 1.5913461538461537, "no_speech_prob": 0.0, "words": [{"start": 278.86, "end": 279.3, "word": " Because", "probability": 0.61279296875}, {"start": 279.3, "end": 279.5, "word": " the", "probability": 0.88720703125}, {"start": 279.5, "end": 279.8, "word": " total", "probability": 0.8740234375}, {"start": 279.8, "end": 280.3, "word": " area", "probability": 0.9111328125}, {"start": 280.3, "end": 280.68, "word": " underneath", "probability": 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{"start": 294.06, "end": 294.28, "word": " 0", "probability": 0.5830078125}, {"start": 294.28, "end": 294.56, "word": ".5.", "probability": 0.99267578125}, {"start": 294.64, "end": 294.78, "word": " So", "probability": 0.9443359375}, {"start": 294.78, "end": 294.96, "word": " this", "probability": 0.87744140625}, {"start": 294.96, "end": 295.1, "word": " is", "probability": 0.92822265625}, {"start": 295.1, "end": 295.36, "word": " your", "probability": 0.8193359375}, {"start": 295.36, "end": 296.3, "word": " answer.", "probability": 0.94482421875}, {"start": 296.82, "end": 296.98, "word": " So", "probability": 0.9541015625}, {"start": 296.98, "end": 297.16, "word": " the", "probability": 0.90087890625}, {"start": 297.16, "end": 297.48, "word": " probability", "probability": 0.95361328125}, {"start": 297.48, "end": 297.76, "word": " of", "probability": 0.96533203125}, {"start": 297.76, "end": 297.88, "word": " Z", "probability": 0.65576171875}, {"start": 297.88, "end": 298.4, "word": " 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Also, it's the same idea here, just find these scores for the two values.", "tokens": [639, 307, 1071, 1365, 281, 14722, 264, 8482, 295, 1783, 5044, 813, 1614, 13, 19, 293, 1649, 13, 2743, 11, 309, 311, 264, 912, 1558, 510, 11, 445, 915, 613, 13444, 337, 264, 732, 4190, 13], "avg_logprob": -0.21558276704839757, "compression_ratio": 1.2222222222222223, "no_speech_prob": 0.0, "words": [{"start": 309.86, "end": 310.3, "word": " This", "probability": 0.61376953125}, {"start": 310.3, "end": 310.48, "word": " is", "probability": 0.9462890625}, {"start": 310.48, "end": 310.82, "word": " another", "probability": 0.9189453125}, {"start": 310.82, "end": 311.36, "word": " example", "probability": 0.96923828125}, {"start": 311.36, "end": 313.88, "word": " to", "probability": 0.67333984375}, {"start": 313.88, "end": 314.4, "word": " compute", "probability": 0.9072265625}, {"start": 314.4, "end": 317.28, "word": " the", "probability": 0.66845703125}, {"start": 317.28, "end": 317.76, "word": " probability", "probability": 0.900390625}, {"start": 317.76, "end": 318.02, "word": " of", "probability": 0.916015625}, {"start": 318.02, "end": 318.24, "word": " X", "probability": 0.78271484375}, {"start": 318.24, "end": 318.76, "word": " greater", "probability": 0.86376953125}, {"start": 318.76, "end": 319.04, "word": " than", "probability": 0.94384765625}, {"start": 319.04, "end": 319.28, "word": " 7", "probability": 0.88134765625}, {"start": 319.28, "end": 319.9, "word": ".4", "probability": 0.978759765625}, {"start": 319.9, "end": 320.2, "word": " and", "probability": 0.85595703125}, {"start": 320.2, "end": 320.48, "word": " 8.", "probability": 0.7958984375}, {"start": 321.6, "end": 322.46, "word": " Also,", "probability": 0.9287109375}, {"start": 322.76, "end": 323.52, "word": " it's", "probability": 0.93115234375}, {"start": 323.52, "end": 323.8, "word": " the", "probability": 0.91748046875}, {"start": 323.8, "end": 324.02, "word": " same", "probability": 0.90869140625}, {"start": 324.02, "end": 324.38, "word": " idea", "probability": 0.92138671875}, {"start": 324.38, "end": 324.8, "word": " here,", "probability": 0.66552734375}, {"start": 325.06, "end": 325.6, "word": " just", "probability": 0.8671875}, {"start": 325.6, "end": 327.34, "word": " find", "probability": 0.818359375}, {"start": 327.34, "end": 328.64, "word": " these", "probability": 0.34130859375}, {"start": 328.64, "end": 329.2, "word": " scores", "probability": 0.751953125}, {"start": 329.2, "end": 329.44, "word": " for", "probability": 0.9462890625}, {"start": 329.44, "end": 329.56, "word": " the", "probability": 0.904296875}, {"start": 329.56, "end": 329.7, "word": " two", "probability": 0.8974609375}, {"start": 329.7, "end": 330.16, "word": " values.", "probability": 0.9677734375}], "temperature": 1.0}, {"id": 13, "seek": 36071, "start": 333.85, "end": 360.71, "text": " L with B of Z greater than minus 0.12 up to 0. Now this red area equals the area below 0. I mean B of Z smaller than 0 minus the probability of Z smaller than minus 0.12. Now by using symmetric probability of the normal distribution, we know that", "tokens": [441, 365, 363, 295, 1176, 5044, 813, 3175, 1958, 13, 4762, 493, 281, 1958, 13, 823, 341, 2182, 1859, 6915, 264, 1859, 2507, 1958, 13, 286, 914, 363, 295, 1176, 4356, 813, 1958, 3175, 264, 8482, 295, 1176, 4356, 813, 3175, 1958, 13, 4762, 13, 823, 538, 1228, 32330, 8482, 295, 264, 2710, 7316, 11, 321, 458, 300], "avg_logprob": -0.219676911325778, "compression_ratio": 1.603896103896104, "no_speech_prob": 0.0, "words": [{"start": 333.85, "end": 334.23, "word": " L", "probability": 0.1065673828125}, {"start": 334.23, "end": 334.53, "word": " with", "probability": 0.8212890625}, {"start": 334.53, "end": 334.79, "word": " B", "probability": 0.458740234375}, {"start": 334.79, "end": 334.95, "word": " of", "probability": 0.8955078125}, {"start": 334.95, "end": 335.13, "word": " Z", "probability": 0.5869140625}, {"start": 335.13, "end": 335.67, "word": " greater", "probability": 0.84765625}, {"start": 335.67, "end": 335.97, "word": " than", "probability": 0.9365234375}, {"start": 335.97, "end": 336.29, "word": " minus", "probability": 0.90087890625}, {"start": 336.29, "end": 336.51, "word": " 0", "probability": 0.642578125}, {"start": 336.51, "end": 336.79, "word": ".12", "probability": 0.984130859375}, {"start": 336.79, "end": 337.01, "word": " up", "probability": 0.947265625}, {"start": 337.01, "end": 337.13, "word": " to", "probability": 0.95703125}, {"start": 337.13, "end": 337.39, "word": " 0.", "probability": 0.56591796875}, {"start": 338.37, "end": 338.67, "word": " Now", "probability": 0.85302734375}, {"start": 338.67, "end": 339.05, "word": " this", "probability": 0.63623046875}, {"start": 339.05, "end": 339.51, "word": " red", "probability": 0.912109375}, {"start": 339.51, "end": 339.97, "word": " area", "probability": 0.87646484375}, {"start": 339.97, "end": 342.47, "word": " equals", "probability": 0.7861328125}, {"start": 342.47, "end": 342.65, "word": " the", "probability": 0.85205078125}, {"start": 342.65, "end": 342.89, "word": " area", "probability": 0.87744140625}, {"start": 342.89, "end": 343.29, "word": " below", "probability": 0.89111328125}, {"start": 343.29, "end": 344.17, "word": " 0.", "probability": 0.69921875}, {"start": 346.21, "end": 346.29, "word": " I", "probability": 0.9658203125}, {"start": 346.29, "end": 346.59, "word": " mean", "probability": 0.962890625}, {"start": 346.59, "end": 346.87, "word": " B", "probability": 0.72412109375}, {"start": 346.87, "end": 347.01, "word": " of", "probability": 0.96533203125}, {"start": 347.01, "end": 347.13, "word": " Z", "probability": 0.94677734375}, {"start": 347.13, "end": 347.51, "word": " smaller", "probability": 0.71484375}, {"start": 347.51, "end": 347.79, "word": " than", "probability": 0.9404296875}, {"start": 347.79, "end": 348.19, "word": " 0", "probability": 0.88427734375}, {"start": 348.19, "end": 349.23, "word": " minus", "probability": 0.83154296875}, {"start": 349.23, "end": 349.53, "word": " the", "probability": 0.767578125}, {"start": 349.53, "end": 349.83, "word": " probability", "probability": 0.90966796875}, {"start": 349.83, "end": 350.11, "word": " of", "probability": 0.8916015625}, {"start": 350.11, "end": 350.33, "word": " Z", "probability": 0.9091796875}, {"start": 350.33, "end": 351.25, "word": " smaller", "probability": 0.83837890625}, {"start": 351.25, "end": 351.57, "word": " than", "probability": 0.93212890625}, {"start": 351.57, "end": 351.93, "word": " minus", "probability": 0.95751953125}, {"start": 351.93, "end": 352.29, "word": " 0", "probability": 0.9599609375}, {"start": 352.29, "end": 353.25, "word": ".12.", "probability": 0.995849609375}, {"start": 354.53, "end": 354.85, "word": " Now", "probability": 0.92578125}, {"start": 354.85, "end": 355.77, "word": " by", "probability": 0.61181640625}, {"start": 355.77, "end": 356.23, "word": " using", "probability": 0.9384765625}, {"start": 356.23, "end": 357.09, "word": " symmetric", "probability": 0.77587890625}, {"start": 357.09, "end": 357.79, "word": " probability", "probability": 0.87255859375}, {"start": 357.79, "end": 358.33, "word": " of", "probability": 0.9638671875}, {"start": 358.33, "end": 358.47, "word": " the", "probability": 0.8876953125}, {"start": 358.47, "end": 358.77, "word": " normal", "probability": 0.873046875}, {"start": 358.77, "end": 359.37, "word": " distribution,", "probability": 0.84814453125}, {"start": 360.13, "end": 360.33, "word": " we", "probability": 0.94140625}, {"start": 360.33, "end": 360.47, "word": " know", "probability": 0.87451171875}, {"start": 360.47, "end": 360.71, "word": " that", "probability": 0.93505859375}], "temperature": 1.0}, {"id": 14, "seek": 38574, "start": 361.12, "end": 385.74, "text": " The probability of Z smaller than minus 0.12 equals the probability of Z greater than 0.12. Because this area, if we have Z smaller than minus 0.12, the area to the left, that equals the area to the right of the same point, because of symmetry. And finally, you will end with this result.", "tokens": [440, 8482, 295, 1176, 4356, 813, 3175, 1958, 13, 4762, 6915, 264, 8482, 295, 1176, 5044, 813, 1958, 13, 4762, 13, 1436, 341, 1859, 11, 498, 321, 362, 1176, 4356, 813, 3175, 1958, 13, 4762, 11, 264, 1859, 281, 264, 1411, 11, 300, 6915, 264, 1859, 281, 264, 558, 295, 264, 912, 935, 11, 570, 295, 25440, 13, 400, 2721, 11, 291, 486, 917, 365, 341, 1874, 13], "avg_logprob": -0.23154439010481903, "compression_ratio": 1.7409638554216869, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 361.12, "end": 361.5, "word": " The", "probability": 0.322265625}, {"start": 361.5, "end": 362.1, "word": " probability", "probability": 0.89990234375}, {"start": 362.1, "end": 362.46, "word": " of", "probability": 0.94580078125}, {"start": 362.46, "end": 362.72, "word": " Z", "probability": 0.5146484375}, {"start": 362.72, "end": 363.32, "word": " smaller", "probability": 0.494873046875}, {"start": 363.32, "end": 363.66, "word": " than", 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It's vice versa. In the previous questions, we were asking about find the probability, for example, if X is smaller than a certain number. Now suppose this probability is given, and we are looking to find this value. 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So if the probability is given less than 0.5, then your z-score should be negative, otherwise should be positive. So you have to be careful in this case. Now look at this example. Let x represent the time it takes in seconds.", "tokens": [583, 498, 710, 12, 4417, 418, 307, 3671, 11, 286, 478, 2597, 11, 498, 710, 12, 4417, 418, 307, 3671, 11, 550, 264, 8482, 820, 312, 4356, 813, 1958, 13, 20, 13, 407, 498, 264, 8482, 307, 2212, 1570, 813, 1958, 13, 20, 11, 550, 428, 710, 12, 4417, 418, 820, 312, 3671, 11, 5911, 820, 312, 3353, 13, 407, 291, 362, 281, 312, 5026, 294, 341, 1389, 13, 823, 574, 412, 341, 1365, 13, 961, 2031, 2906, 264, 565, 309, 2516, 294, 3949, 13], "avg_logprob": -0.1478924385683481, "compression_ratio": 1.6984924623115578, "no_speech_prob": 0.0, "words": [{"start": 605.54, "end": 605.9, "word": " But", "probability": 0.66552734375}, {"start": 605.9, "end": 606.1, "word": " if", "probability": 0.90185546875}, {"start": 606.1, "end": 606.28, "word": " z", "probability": 0.55224609375}, {"start": 606.28, "end": 606.68, "word": "-score", "probability": 0.751953125}, {"start": 606.68, "end": 606.94, "word": " is", "probability": 0.94921875}, {"start": 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image file from the internet. The same example as we did before. And here we assume that x is normal distribution with mean of 8 and standard deviation of 5. Now, let's see how can we find the value of x such that 20% of download times are smaller than x.", "tokens": [281, 5484, 364, 3256, 3991, 490, 264, 4705, 13, 440, 912, 1365, 382, 321, 630, 949, 13, 400, 510, 321, 6552, 300, 2031, 307, 2710, 7316, 365, 914, 295, 1649, 293, 3832, 25163, 295, 1025, 13, 823, 11, 718, 311, 536, 577, 393, 321, 915, 264, 2158, 295, 2031, 1270, 300, 945, 4, 295, 5484, 1413, 366, 4356, 813, 2031, 13], "avg_logprob": -0.1678427397724121, "compression_ratio": 1.433862433862434, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 633.71, "end": 633.97, "word": " to", "probability": 0.39404296875}, {"start": 633.97, "end": 634.45, "word": " download", "probability": 0.96142578125}, {"start": 634.45, "end": 634.67, "word": " an", "probability": 0.8779296875}, {"start": 634.67, "end": 635.05, "word": " image", "probability": 0.9228515625}, {"start": 635.05, "end": 635.77, "word": " file", "probability": 0.525390625}, {"start": 635.77, "end": 636.05, "word": 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Also, always the probability is between 0 and 1. So, the probability here is 20%. In this case, your z-score should be negative. Because 20% is more than 0.5. So, z-score should be in this side, in the left side. So, again, he asks about finding x-value such that 20%.", "tokens": [407, 11, 341, 8482, 307, 257, 14135, 13, 2743, 11, 1009, 264, 8482, 307, 1296, 1958, 293, 502, 13, 407, 11, 264, 8482, 510, 307, 945, 6856, 682, 341, 1389, 11, 428, 710, 12, 4417, 418, 820, 312, 3671, 13, 1436, 945, 4, 307, 544, 813, 1958, 13, 20, 13, 407, 11, 710, 12, 4417, 418, 820, 312, 294, 341, 1252, 11, 294, 264, 1411, 1252, 13, 407, 11, 797, 11, 415, 8962, 466, 5006, 2031, 12, 29155, 1270, 300, 945, 6856], "avg_logprob": -0.2658132645021002, "compression_ratio": 1.622340425531915, "no_speech_prob": 0.0, "words": [{"start": 661.06, "end": 661.34, "word": " So,", "probability": 0.497802734375}, {"start": 661.56, "end": 661.56, "word": " this", "probability": 0.80908203125}, {"start": 661.56, "end": 661.94, "word": " probability", "probability": 0.919921875}, {"start": 661.94, "end": 662.24, "word": " is", "probability": 0.919921875}, {"start": 662.24, "end": 662.36, "word": " a", "probability": 0.87109375}, 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Find a z-score corresponding to the probability of 0.2. The approximate one, the near value, I mean, to the 0.2 is 0.2005. Sometimes you have the exact value from the table you have, but most of the time you don't have it. 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The closest value to 0.2 is 0.2005. Now, the corresponding value to this probability is minus 0.8 all the way up to 4. So your z-score is negative 0.84. So this is the first step. Any question? Again.", "tokens": [407, 510, 11, 321, 366, 1237, 337, 1958, 13, 17, 13, 440, 13699, 2158, 281, 1958, 13, 17, 307, 1958, 13, 7629, 20, 13, 823, 11, 264, 11760, 2158, 281, 341, 8482, 307, 3175, 1958, 13, 23, 439, 264, 636, 493, 281, 1017, 13, 407, 428, 710, 12, 4417, 418, 307, 3671, 1958, 13, 25494, 13, 407, 341, 307, 264, 700, 1823, 13, 2639, 1168, 30, 3764, 13], "avg_logprob": -0.1692708350610042, "compression_ratio": 1.3928571428571428, "no_speech_prob": 0.0, "words": [{"start": 742.6, "end": 742.86, "word": " So", "probability": 0.6748046875}, {"start": 742.86, "end": 743.08, "word": " here,", "probability": 0.65869140625}, {"start": 743.16, "end": 743.3, "word": " we", "probability": 0.95263671875}, {"start": 743.3, "end": 743.46, "word": " are", "probability": 0.9130859375}, {"start": 743.46, "end": 743.7, "word": " looking", "probability": 0.919921875}, {"start": 743.7, "end": 743.98, "word": " for", "probability": 0.9541015625}, {"start": 743.98, "end": 744.32, "word": " 0", "probability": 0.55908203125}, {"start": 744.32, "end": 744.68, "word": ".2.", "probability": 0.979248046875}, {"start": 745.38, "end": 745.84, "word": " The", "probability": 0.8779296875}, {"start": 745.84, "end": 746.22, "word": " closest", "probability": 0.91943359375}, {"start": 746.22, "end": 746.7, "word": " value", "probability": 0.95068359375}, {"start": 746.7, "end": 746.92, "word": " to", "probability": 0.6279296875}, {"start": 746.92, "end": 747.16, "word": " 0", "probability": 0.81396484375}, {"start": 747.16, "end": 747.28, "word": ".2", "probability": 0.98486328125}, {"start": 747.28, "end": 747.5, "word": " is", "probability": 0.92138671875}, {"start": 747.5, "end": 747.76, "word": " 0", "probability": 0.953125}, {"start": 747.76, "end": 748.74, "word": ".2005.", "probability": 0.7913411458333334}, {"start": 749.54, "end": 750.22, "word": " Now,", "probability": 0.95166015625}, {"start": 750.36, "end": 750.66, "word": " the", "probability": 0.91845703125}, {"start": 750.66, "end": 751.16, "word": " corresponding", "probability": 0.83935546875}, {"start": 751.16, "end": 751.46, "word": " value", "probability": 0.96826171875}, {"start": 751.46, "end": 751.7, "word": " to", "probability": 0.921875}, {"start": 751.7, "end": 752.16, "word": " this", "probability": 0.93359375}, {"start": 752.16, "end": 752.88, "word": " probability", "probability": 0.9609375}, {"start": 752.88, "end": 753.98, "word": " is", "probability": 0.9208984375}, {"start": 753.98, "end": 754.34, "word": " minus", "probability": 0.486572265625}, {"start": 754.34, "end": 754.72, "word": " 0", "probability": 0.88623046875}, {"start": 754.72, "end": 755.26, "word": ".8", "probability": 0.9892578125}, {"start": 755.26, "end": 755.94, "word": " all", "probability": 0.7705078125}, {"start": 755.94, "end": 756.12, "word": " the", "probability": 0.919921875}, {"start": 756.12, "end": 756.32, "word": " way", "probability": 0.95263671875}, {"start": 756.32, "end": 756.66, "word": " up", "probability": 0.96044921875}, {"start": 756.66, "end": 756.96, "word": " to", "probability": 0.96728515625}, {"start": 756.96, "end": 757.68, "word": " 4.", "probability": 0.71337890625}, {"start": 758.36, "end": 758.84, "word": " So", "probability": 0.9599609375}, {"start": 758.84, "end": 759.42, "word": " your", "probability": 0.7001953125}, {"start": 759.42, "end": 759.62, "word": " z", "probability": 0.43896484375}, {"start": 759.62, "end": 759.92, "word": "-score", "probability": 0.8284505208333334}, {"start": 759.92, "end": 760.12, "word": " is", "probability": 0.93896484375}, {"start": 760.12, "end": 760.54, "word": " negative", "probability": 0.875}, {"start": 760.54, "end": 761.72, "word": " 0", "probability": 0.98095703125}, {"start": 761.72, "end": 762.36, "word": ".84.", "probability": 0.93994140625}, {"start": 763.28, "end": 763.86, "word": " So", "probability": 0.951171875}, {"start": 763.86, "end": 764.04, "word": " this", "probability": 0.90673828125}, {"start": 764.04, "end": 764.16, "word": " is", "probability": 0.93603515625}, {"start": 764.16, "end": 764.26, "word": " the", "probability": 0.91748046875}, {"start": 764.26, "end": 764.5, "word": " first", "probability": 0.88330078125}, {"start": 764.5, "end": 764.84, "word": " step.", "probability": 0.9052734375}, {"start": 767.12, "end": 767.84, "word": " Any", "probability": 0.8974609375}, {"start": 767.84, "end": 768.2, "word": " question?", "probability": 0.64306640625}, {"start": 770.4, "end": 771.12, "word": " Again.", "probability": 0.9501953125}], "temperature": 1.0}, {"id": 29, "seek": 79751, "start": 773.95, "end": 797.51, "text": " Now if we just go back to this equation, z equals x minus mu over sigma. 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So here we need to find the value of z first. Go back to the normal table you have. This is the normal table. We are looking for minus 0.2.", "tokens": [823, 264, 8482, 307, 1570, 813, 1958, 13, 20, 11, 370, 428, 710, 12, 4417, 418, 820, 312, 322, 264, 1411, 1252, 13, 407, 510, 321, 643, 281, 915, 264, 2158, 295, 710, 700, 13, 1037, 646, 281, 264, 2710, 3199, 291, 362, 13, 639, 307, 264, 2710, 3199, 13, 492, 366, 1237, 337, 3175, 1958, 13, 17, 13], "avg_logprob": -0.16549479762713115, "compression_ratio": 1.4050632911392404, "no_speech_prob": 0.0, "words": [{"start": 830.74, "end": 831.0, "word": " Now", "probability": 0.5830078125}, {"start": 831.0, "end": 831.16, "word": " the", "probability": 0.63623046875}, {"start": 831.16, "end": 831.46, "word": " probability", "probability": 0.91015625}, {"start": 831.46, "end": 831.8, "word": " is", "probability": 0.91162109375}, {"start": 831.8, "end": 832.0, "word": " less", "probability": 0.880859375}, {"start": 832.0, "end": 832.16, "word": " than", "probability": 0.92822265625}, {"start": 832.16, "end": 832.36, "word": " 0", "probability": 0.65478515625}, {"start": 832.36, "end": 832.78, "word": ".5,", "probability": 0.9873046875}, {"start": 832.98, "end": 833.3, "word": " so", "probability": 0.87353515625}, {"start": 833.3, "end": 833.9, "word": " your", "probability": 0.81884765625}, {"start": 833.9, "end": 834.08, "word": " z", "probability": 0.6748046875}, {"start": 834.08, "end": 834.4, "word": "-score", "probability": 0.7716471354166666}, {"start": 834.4, "end": 834.68, "word": " should", "probability": 0.96435546875}, {"start": 834.68, "end": 834.94, "word": " be", "probability": 0.94775390625}, {"start": 834.94, "end": 835.22, "word": " on", "probability": 0.5009765625}, {"start": 835.22, "end": 835.34, "word": " the", "probability": 0.91259765625}, {"start": 835.34, "end": 835.52, "word": " left", "probability": 0.94384765625}, {"start": 835.52, "end": 835.96, "word": " side.", "probability": 0.77734375}, {"start": 836.54, "end": 836.7, "word": " So", "probability": 0.88916015625}, {"start": 836.7, "end": 836.98, "word": " here", "probability": 0.76904296875}, {"start": 836.98, "end": 837.48, "word": " we", "probability": 0.59326171875}, {"start": 837.48, "end": 837.78, "word": " need", "probability": 0.91455078125}, {"start": 837.78, "end": 838.36, "word": " to", "probability": 0.95068359375}, {"start": 838.36, "end": 838.88, "word": " find", "probability": 0.89599609375}, {"start": 838.88, "end": 839.08, "word": " the", "probability": 0.9072265625}, {"start": 839.08, "end": 839.5, "word": " value", "probability": 0.9765625}, {"start": 839.5, "end": 839.78, "word": " of", "probability": 0.96533203125}, {"start": 839.78, "end": 839.98, "word": " z", "probability": 0.90869140625}, {"start": 839.98, "end": 840.4, "word": " first.", "probability": 0.8076171875}, {"start": 842.3, "end": 842.68, "word": " Go", "probability": 0.9306640625}, {"start": 842.68, "end": 843.02, "word": " back", "probability": 0.876953125}, {"start": 843.02, "end": 843.5, "word": " to", "probability": 0.9521484375}, {"start": 843.5, "end": 843.66, "word": " the", "probability": 0.90771484375}, {"start": 843.66, "end": 843.92, "word": " normal", "probability": 0.88623046875}, {"start": 843.92, "end": 844.22, "word": " table", "probability": 0.865234375}, {"start": 844.22, "end": 844.42, "word": " you", "probability": 0.79443359375}, {"start": 844.42, "end": 844.64, "word": " have.", "probability": 0.8369140625}, {"start": 847.68, "end": 848.02, "word": " This", "probability": 0.845703125}, {"start": 848.02, "end": 848.12, "word": " is", "probability": 0.95166015625}, {"start": 848.12, "end": 848.24, "word": " the", "probability": 0.837890625}, {"start": 848.24, "end": 848.48, "word": " normal", "probability": 0.88525390625}, {"start": 848.48, "end": 848.86, "word": " table.", "probability": 0.89453125}, {"start": 856.8, "end": 857.36, "word": " We", "probability": 0.93994140625}, {"start": 857.36, "end": 857.5, "word": " are", "probability": 0.88427734375}, {"start": 857.5, "end": 857.78, "word": " looking", "probability": 0.91357421875}, {"start": 857.78, "end": 858.22, "word": " for", "probability": 0.95166015625}, {"start": 858.22, "end": 858.62, "word": " minus", "probability": 0.61376953125}, {"start": 858.62, "end": 858.88, "word": " 0", "probability": 0.94189453125}, {"start": 858.88, "end": 859.18, "word": ".2.", "probability": 0.992431640625}], "temperature": 1.0}, {"id": 32, "seek": 87475, "start": 860.27, "end": 874.75, "text": " I'm sorry, we are looking for 0.2. So the closest value to 0.2 is this one, 0.2005. So this is the closest value.", "tokens": [286, 478, 2597, 11, 321, 366, 1237, 337, 1958, 13, 17, 13, 407, 264, 13699, 2158, 281, 1958, 13, 17, 307, 341, 472, 11, 1958, 13, 7629, 20, 13, 407, 341, 307, 264, 13699, 2158, 13], "avg_logprob": -0.17694257401131294, "compression_ratio": 1.2527472527472527, "no_speech_prob": 7.152557373046875e-07, "words": [{"start": 860.27, "end": 860.83, "word": " I'm", "probability": 0.6453857421875}, {"start": 860.83, "end": 861.03, "word": " sorry,", "probability": 0.88037109375}, {"start": 861.11, "end": 861.15, "word": " we", "probability": 0.9404296875}, {"start": 861.15, "end": 861.25, "word": " are", "probability": 0.7265625}, {"start": 861.25, "end": 861.47, "word": " looking", "probability": 0.91650390625}, {"start": 861.47, "end": 861.91, "word": " for", "probability": 0.9521484375}, {"start": 861.91, "end": 862.81, "word": " 0", "probability": 0.331298828125}, {"start": 862.81, "end": 863.21, "word": ".2.", "probability": 0.96484375}, {"start": 865.23, "end": 865.91, "word": " So", "probability": 0.91650390625}, {"start": 865.91, "end": 866.11, "word": " the", "probability": 0.6240234375}, {"start": 866.11, "end": 866.45, "word": " closest", "probability": 0.92333984375}, {"start": 866.45, "end": 866.95, "word": " value", "probability": 0.97265625}, {"start": 866.95, "end": 868.13, "word": " to", "probability": 0.91015625}, {"start": 868.13, "end": 868.45, "word": " 0", "probability": 0.90673828125}, {"start": 868.45, "end": 868.69, "word": ".2", "probability": 0.993896484375}, {"start": 868.69, "end": 868.91, "word": " is", "probability": 0.92529296875}, {"start": 868.91, "end": 869.11, "word": " this", "probability": 0.908203125}, {"start": 869.11, "end": 869.41, "word": " one,", "probability": 0.8994140625}, {"start": 870.55, "end": 870.83, "word": " 0", "probability": 0.9716796875}, {"start": 870.83, "end": 871.97, "word": ".2005.", "probability": 0.9046223958333334}, {"start": 872.91, "end": 873.37, "word": " So", "probability": 0.94677734375}, {"start": 873.37, "end": 873.55, "word": " this", "probability": 0.76611328125}, {"start": 873.55, "end": 873.65, "word": " is", "probability": 0.9384765625}, {"start": 873.65, "end": 873.85, "word": " the", "probability": 0.89892578125}, {"start": 873.85, "end": 874.29, "word": " closest", "probability": 0.94677734375}, {"start": 874.29, "end": 874.75, "word": " value.", "probability": 0.9755859375}], "temperature": 1.0}, {"id": 33, "seek": 91755, "start": 889.63, "end": 917.55, "text": " So the exact answer is sometimes not given. So the approximate one, minus 0.8, all the way up to 4. So z-score minus 0.8. Any question? So the value of z-score is minus 0.84. So my corresponding x-value equals", "tokens": [407, 264, 1900, 1867, 307, 2171, 406, 2212, 13, 407, 264, 30874, 472, 11, 3175, 1958, 13, 23, 11, 439, 264, 636, 493, 281, 1017, 13, 407, 710, 12, 4417, 418, 3175, 1958, 13, 23, 13, 2639, 1168, 30, 407, 264, 2158, 295, 710, 12, 4417, 418, 307, 3175, 1958, 13, 25494, 13, 407, 452, 11760, 2031, 12, 29155, 6915], "avg_logprob": -0.21273052594700798, "compression_ratio": 1.4685314685314685, "no_speech_prob": 0.0, "words": [{"start": 889.63, "end": 890.11, "word": " So", "probability": 0.62890625}, {"start": 890.11, "end": 890.61, "word": " the", "probability": 0.6240234375}, {"start": 890.61, "end": 891.05, "word": " exact", "probability": 0.94677734375}, {"start": 891.05, "end": 891.71, "word": " answer", "probability": 0.94921875}, {"start": 891.71, "end": 892.15, "word": " is", "probability": 0.344970703125}, {"start": 892.15, "end": 892.61, "word": " sometimes", "probability": 0.93505859375}, {"start": 892.61, "end": 892.97, "word": " not", "probability": 0.93798828125}, {"start": 892.97, "end": 893.29, "word": " given.", "probability": 0.875}, {"start": 894.05, "end": 894.31, "word": " So", "probability": 0.93896484375}, {"start": 894.31, "end": 894.47, "word": " the", "probability": 0.849609375}, {"start": 894.47, "end": 894.85, "word": " approximate", "probability": 0.857421875}, {"start": 894.85, "end": 895.23, "word": " one,", "probability": 0.93115234375}, {"start": 895.47, "end": 895.73, "word": " minus", "probability": 0.76708984375}, {"start": 895.73, "end": 896.01, "word": " 0", "probability": 0.60693359375}, {"start": 896.01, "end": 896.43, "word": ".8,", "probability": 0.99267578125}, {"start": 896.59, "end": 896.99, "word": " all", "probability": 0.9453125}, {"start": 896.99, "end": 897.15, "word": " the", "probability": 0.91357421875}, {"start": 897.15, "end": 897.31, "word": " way", "probability": 0.95703125}, {"start": 897.31, "end": 897.55, "word": " up", "probability": 0.95947265625}, {"start": 897.55, "end": 898.25, "word": " to", "probability": 0.9677734375}, {"start": 898.25, "end": 899.19, "word": " 4.", "probability": 0.587890625}, {"start": 900.03, "end": 900.51, "word": " So", "probability": 0.93408203125}, {"start": 900.51, "end": 900.81, "word": " z", "probability": 0.382568359375}, {"start": 900.81, "end": 901.19, "word": "-score", "probability": 0.76318359375}, {"start": 901.19, "end": 903.59, "word": " minus", "probability": 0.3193359375}, {"start": 903.59, "end": 903.95, "word": " 0", "probability": 0.978515625}, {"start": 903.95, "end": 904.27, "word": ".8.", "probability": 0.978759765625}, {"start": 905.53, "end": 906.03, "word": " Any", "probability": 0.892578125}, {"start": 906.03, "end": 906.41, "word": " question?", "probability": 0.6435546875}, {"start": 910.33, "end": 911.09, "word": " So", "probability": 0.84912109375}, {"start": 911.09, "end": 911.75, "word": " the", "probability": 0.8349609375}, {"start": 911.75, "end": 911.99, "word": " value", "probability": 0.9677734375}, {"start": 911.99, "end": 912.11, "word": " of", "probability": 0.962890625}, {"start": 912.11, "end": 912.27, "word": " z", "probability": 0.751953125}, {"start": 912.27, "end": 912.53, "word": "-score", "probability": 0.9235026041666666}, {"start": 912.53, "end": 912.75, "word": " is", "probability": 0.94091796875}, {"start": 912.75, "end": 913.11, "word": " minus", "probability": 0.97314453125}, {"start": 913.11, "end": 913.37, "word": " 0", "probability": 0.99609375}, {"start": 913.37, "end": 913.71, "word": ".84.", "probability": 0.994384765625}, {"start": 914.65, "end": 914.93, "word": " So", "probability": 0.96240234375}, {"start": 914.93, "end": 915.33, "word": " my", "probability": 0.9609375}, {"start": 915.33, "end": 916.23, "word": " corresponding", "probability": 0.67822265625}, {"start": 916.23, "end": 916.73, "word": " x", "probability": 0.95361328125}, {"start": 916.73, "end": 917.03, "word": "-value", "probability": 0.709228515625}, {"start": 917.03, "end": 917.55, "word": " equals", "probability": 0.93359375}], "temperature": 1.0}, {"id": 34, "seek": 94535, "start": 919.15, "end": 945.35, "text": " X equal mu. The mu is given as A plus Z is minus 0.84 times sigma. Sigma is 5. You will end with 3.8. So this means the probability of X less than 3.8.", "tokens": [1783, 2681, 2992, 13, 440, 2992, 307, 2212, 382, 316, 1804, 1176, 307, 3175, 1958, 13, 25494, 1413, 12771, 13, 36595, 307, 1025, 13, 509, 486, 917, 365, 805, 13, 23, 13, 407, 341, 1355, 264, 8482, 295, 1783, 1570, 813, 805, 13, 23, 13], "avg_logprob": -0.26528534034024115, "compression_ratio": 1.2459016393442623, "no_speech_prob": 0.0, "words": [{"start": 919.15, "end": 919.81, "word": " X", "probability": 0.3984375}, {"start": 919.81, "end": 920.13, "word": " equal", "probability": 0.393798828125}, {"start": 920.13, "end": 920.45, "word": " mu.", "probability": 0.457763671875}, {"start": 920.95, "end": 921.17, "word": " The", "probability": 0.57177734375}, {"start": 921.17, "end": 921.29, "word": " mu", "probability": 0.90576171875}, {"start": 921.29, "end": 921.43, "word": " is", "probability": 0.65576171875}, {"start": 921.43, "end": 921.71, "word": " given", "probability": 0.88330078125}, {"start": 921.71, "end": 922.81, "word": " as", "probability": 0.6923828125}, {"start": 922.81, "end": 923.09, "word": " A", "probability": 0.7060546875}, {"start": 923.09, "end": 924.25, "word": " plus", "probability": 0.69482421875}, {"start": 924.25, "end": 926.87, "word": " Z", "probability": 0.42724609375}, {"start": 926.87, "end": 927.17, "word": " is", "probability": 0.892578125}, {"start": 927.17, "end": 927.61, "word": " minus", "probability": 0.95556640625}, {"start": 927.61, "end": 927.95, "word": " 0", "probability": 0.77734375}, {"start": 927.95, "end": 929.13, "word": ".84", "probability": 0.992431640625}, {"start": 929.13, "end": 930.49, "word": " times", "probability": 0.580078125}, {"start": 930.49, "end": 930.95, "word": " sigma.", "probability": 0.8447265625}, {"start": 931.33, "end": 931.77, "word": " Sigma", "probability": 0.8681640625}, {"start": 931.77, "end": 932.07, "word": " is", "probability": 0.95068359375}, {"start": 932.07, "end": 932.35, "word": " 5.", "probability": 0.66064453125}, {"start": 934.31, "end": 934.65, "word": " You", "probability": 0.76123046875}, {"start": 934.65, "end": 934.81, "word": " will", "probability": 0.86669921875}, {"start": 934.81, "end": 935.19, "word": " end", "probability": 0.8916015625}, {"start": 935.19, "end": 935.97, "word": " with", "probability": 0.80615234375}, {"start": 935.97, "end": 936.27, "word": " 3", "probability": 0.892578125}, {"start": 936.27, "end": 936.63, "word": ".8.", "probability": 0.9482421875}, {"start": 938.65, "end": 939.37, "word": " So", "probability": 0.95556640625}, {"start": 939.37, "end": 939.63, "word": " this", "probability": 0.85009765625}, {"start": 939.63, "end": 940.03, "word": " means", "probability": 0.9326171875}, {"start": 940.03, "end": 942.15, "word": " the", "probability": 0.76904296875}, {"start": 942.15, "end": 942.51, "word": " probability", "probability": 0.82177734375}, {"start": 942.51, "end": 942.79, "word": " of", "probability": 0.95654296875}, {"start": 942.79, "end": 943.15, "word": " X", "probability": 0.91162109375}, {"start": 943.15, "end": 944.29, "word": " less", "probability": 0.931640625}, {"start": 944.29, "end": 944.53, "word": " than", "probability": 0.94140625}, {"start": 944.53, "end": 944.81, "word": " 3", "probability": 0.98828125}, {"start": 944.81, "end": 945.35, "word": ".8.", "probability": 0.998779296875}], "temperature": 1.0}, {"id": 35, "seek": 97525, "start": 947.01, "end": 975.25, "text": " Equal point. Exactly, equal point. So in this case, the probability is given, which is 0.20. And we ask about what's the value of x in this case. So the first step was find the z-score. Then use this value. 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That's the backward normal calculations. Let's do one problem from the practice, which is number 18.", "tokens": [281, 915, 264, 11760, 1783, 6175, 13, 663, 311, 264, 23897, 2710, 20448, 13, 961, 311, 360, 472, 1154, 490, 264, 3124, 11, 597, 307, 1230, 2443, 13], "avg_logprob": -0.23208512519967966, "compression_ratio": 1.1929824561403508, "no_speech_prob": 0.0, "words": [{"start": 977.38, "end": 977.62, "word": " to", "probability": 0.476806640625}, {"start": 977.62, "end": 977.88, "word": " find", "probability": 0.890625}, {"start": 977.88, "end": 978.06, "word": " the", "probability": 0.640625}, {"start": 978.06, "end": 978.52, "word": " corresponding", "probability": 0.666015625}, {"start": 978.52, "end": 978.96, "word": " X", "probability": 0.5888671875}, {"start": 978.96, "end": 979.38, "word": " score.", "probability": 0.51318359375}, {"start": 981.5, "end": 982.22, "word": " That's", "probability": 0.76220703125}, {"start": 982.22, "end": 982.52, "word": " the", "probability": 0.86279296875}, {"start": 982.52, "end": 983.1, "word": " backward", "probability": 0.91064453125}, {"start": 983.1, "end": 984.08, "word": " normal", "probability": 0.7548828125}, {"start": 984.08, "end": 985.24, "word": " calculations.", "probability": 0.71533203125}, {"start": 988.82, "end": 989.82, "word": " Let's", "probability": 0.968505859375}, {"start": 989.82, "end": 990.08, "word": " do", "probability": 0.90625}, {"start": 990.08, "end": 990.98, "word": " one", "probability": 0.92626953125}, {"start": 990.98, "end": 991.36, "word": " problem", "probability": 0.89013671875}, {"start": 991.36, "end": 991.68, "word": " from", "probability": 0.88671875}, {"start": 991.68, "end": 991.84, "word": " the", "probability": 0.8359375}, {"start": 991.84, "end": 992.38, "word": " practice,", "probability": 0.93310546875}, {"start": 994.18, "end": 994.76, "word": " which", "probability": 0.951171875}, {"start": 994.76, "end": 994.92, "word": " is", "probability": 0.947265625}, {"start": 994.92, "end": 995.18, "word": " number", "probability": 0.86767578125}, {"start": 995.18, "end": 996.82, "word": " 18.", "probability": 0.9443359375}], "temperature": 1.0}, {"id": 37, "seek": 103471, "start": 1013.39, "end": 1034.71, "text": " Is it clear? The owners of a fish market determined that the average weight for a catfish", "tokens": [1119, 309, 1850, 30, 440, 7710, 295, 257, 3506, 2142, 9540, 300, 264, 4274, 3364, 337, 257, 3857, 11608], "avg_logprob": -0.21464843973517417, "compression_ratio": 1.125, "no_speech_prob": 0.0, "words": [{"start": 1013.3900000000001, "end": 1014.69, "word": " Is", "probability": 0.6357421875}, {"start": 1014.69, "end": 1015.99, "word": " it", "probability": 0.8623046875}, {"start": 1015.99, "end": 1016.31, "word": " clear?", "probability": 0.8837890625}, {"start": 1020.59, "end": 1021.89, "word": " The", "probability": 0.75634765625}, {"start": 1021.89, "end": 1022.51, "word": " owners", "probability": 0.1806640625}, {"start": 1022.51, "end": 1025.03, "word": " of", "probability": 0.95458984375}, {"start": 1025.03, "end": 1025.25, "word": " a", "probability": 0.98486328125}, {"start": 1025.25, "end": 1025.45, "word": " fish", "probability": 0.92529296875}, {"start": 1025.45, "end": 1026.13, "word": " market", "probability": 0.8974609375}, {"start": 1026.13, "end": 1030.31, "word": " determined", "probability": 0.91259765625}, {"start": 1030.31, "end": 1031.77, "word": " that", "probability": 0.923828125}, {"start": 1031.77, "end": 1032.77, "word": " the", "probability": 0.8671875}, {"start": 1032.77, "end": 1033.17, "word": " average", "probability": 0.76318359375}, {"start": 1033.17, "end": 1033.63, "word": " weight", "probability": 0.919921875}, {"start": 1033.63, "end": 1033.97, "word": " for", "probability": 0.70654296875}, {"start": 1033.97, "end": 1034.13, "word": " a", "probability": 0.97998046875}, {"start": 1034.13, "end": 1034.71, "word": " catfish", "probability": 0.945068359375}], "temperature": 1.0}, {"id": 38, "seek": 105915, "start": 1035.67, "end": 1059.15, "text": " is 3.2 So this is the value of the mean 40", "tokens": [307, 805, 13, 17, 407, 341, 307, 264, 2158, 295, 264, 914, 3356], "avg_logprob": -0.3722098299435207, "compression_ratio": 0.9555555555555556, "no_speech_prob": 0.0, "words": [{"start": 1035.67, "end": 1036.13, "word": " is", "probability": 0.468017578125}, {"start": 1036.13, "end": 1036.57, "word": " 3", "probability": 0.76611328125}, {"start": 1036.57, "end": 1037.07, "word": ".2", "probability": 0.940673828125}, {"start": 1037.07, "end": 1040.57, "word": " So", "probability": 0.162109375}, {"start": 1040.57, "end": 1042.47, "word": " this", "probability": 0.80615234375}, {"start": 1042.47, "end": 1042.57, "word": " is", "probability": 0.9365234375}, {"start": 1042.57, "end": 1047.23, "word": " the", "probability": 0.8798828125}, {"start": 1047.23, "end": 1047.25, "word": " value", "probability": 0.974609375}, {"start": 1047.25, "end": 1047.27, "word": " of", "probability": 0.9658203125}, {"start": 1047.27, "end": 1047.99, "word": " the", "probability": 0.7744140625}, {"start": 1047.99, "end": 1047.99, "word": " mean", "probability": 0.89697265625}, {"start": 1047.99, "end": 1059.15, "word": " 40", "probability": 0.3701171875}], "temperature": 1.0}, {"id": 39, "seek": 109163, "start": 1068.31, "end": 1091.63, "text": " So again, the owner of a fish market determined that the average weight for a catfish is 3.2 pounds. So the mean is 3.2 with a standard deviation of 0.8.", "tokens": [407, 797, 11, 264, 7289, 295, 257, 3506, 2142, 9540, 300, 264, 4274, 3364, 337, 257, 3857, 11608, 307, 805, 13, 17, 8319, 13, 407, 264, 914, 307, 805, 13, 17, 365, 257, 3832, 25163, 295, 1958, 13, 23, 13], "avg_logprob": -0.1697789605070905, "compression_ratio": 1.305084745762712, "no_speech_prob": 0.0, "words": [{"start": 1068.3099999999997, "end": 1069.1899999999998, "word": " So", "probability": 0.471923828125}, {"start": 1069.1899999999998, "end": 1070.07, "word": " again,", "probability": 0.8369140625}, {"start": 1071.31, "end": 1073.31, "word": " the", "probability": 0.82861328125}, {"start": 1073.31, "end": 1073.71, "word": " owner", "probability": 0.5107421875}, {"start": 1073.71, "end": 1074.03, "word": " of", "probability": 0.97412109375}, {"start": 1074.03, "end": 1074.13, "word": " a", "probability": 0.9462890625}, {"start": 1074.13, "end": 1074.33, "word": " fish", "probability": 0.92919921875}, {"start": 1074.33, "end": 1074.97, "word": " market", "probability": 0.88720703125}, {"start": 1074.97, "end": 1077.01, "word": " determined", "probability": 0.89306640625}, {"start": 1077.01, "end": 1077.59, "word": " that", "probability": 0.908203125}, {"start": 1077.59, "end": 1077.75, "word": " the", "probability": 0.89013671875}, {"start": 1077.75, "end": 1078.07, "word": " average", "probability": 0.7744140625}, {"start": 1078.07, "end": 1078.57, "word": " weight", "probability": 0.91845703125}, {"start": 1078.57, "end": 1080.13, "word": " for", "probability": 0.71484375}, {"start": 1080.13, "end": 1080.33, "word": " a", "probability": 0.90771484375}, {"start": 1080.33, "end": 1080.85, "word": " catfish", "probability": 0.933349609375}, {"start": 1080.85, "end": 1081.17, "word": " is", "probability": 0.94970703125}, {"start": 1081.17, "end": 1081.43, "word": " 3", "probability": 0.951171875}, {"start": 1081.43, "end": 1082.35, "word": ".2", "probability": 0.98388671875}, {"start": 1082.35, "end": 1082.77, "word": " pounds.", "probability": 0.86181640625}, {"start": 1083.37, "end": 1083.57, "word": " So", "probability": 0.8193359375}, {"start": 1083.57, "end": 1083.73, "word": " the", "probability": 0.845703125}, {"start": 1083.73, "end": 1084.01, "word": " mean", "probability": 0.9443359375}, {"start": 1084.01, "end": 1086.35, "word": " is", "probability": 0.8916015625}, {"start": 1086.35, "end": 1086.57, "word": " 3", "probability": 0.9892578125}, {"start": 1086.57, "end": 1087.03, "word": ".2", "probability": 0.9921875}, {"start": 1087.03, "end": 1088.43, "word": " with", "probability": 0.451416015625}, {"start": 1088.43, "end": 1088.73, "word": " a", "probability": 0.95068359375}, {"start": 1088.73, "end": 1088.95, "word": " standard", "probability": 0.9033203125}, {"start": 1088.95, "end": 1089.39, "word": " deviation", "probability": 0.923828125}, {"start": 1089.39, "end": 1089.95, "word": " of", "probability": 0.9716796875}, {"start": 1089.95, "end": 1091.19, "word": " 0", "probability": 0.69091796875}, {"start": 1091.19, "end": 1091.63, "word": ".8.", "probability": 0.993408203125}], "temperature": 1.0}, {"id": 40, "seek": 111706, "start": 1092.1, "end": 1117.06, "text": " So sigma is 0.8. Now, assuming the weights of catfish are normally distributed. In this case, you ask about what's the probability that a randomly selected catfish will weigh more than 4.4. So what's the probability of X", "tokens": [407, 12771, 307, 1958, 13, 23, 13, 823, 11, 11926, 264, 17443, 295, 3857, 11608, 366, 5646, 12631, 13, 682, 341, 1389, 11, 291, 1029, 466, 437, 311, 264, 8482, 300, 257, 16979, 8209, 3857, 11608, 486, 13843, 544, 813, 1017, 13, 19, 13, 407, 437, 311, 264, 8482, 295, 1783], "avg_logprob": -0.21259014480389082, "compression_ratio": 1.4258064516129032, "no_speech_prob": 0.0, "words": [{"start": 1092.1, "end": 1092.64, "word": " So", "probability": 0.239990234375}, {"start": 1092.64, "end": 1092.78, "word": " sigma", "probability": 0.435791015625}, {"start": 1092.78, "end": 1094.92, "word": " is", "probability": 0.89501953125}, {"start": 1094.92, "end": 1095.12, "word": " 0", "probability": 0.467041015625}, {"start": 1095.12, "end": 1095.36, "word": ".8.", "probability": 0.976318359375}, {"start": 1097.1, "end": 1097.68, "word": " Now,", "probability": 0.92626953125}, {"start": 1097.84, "end": 1098.28, "word": " assuming", "probability": 0.89111328125}, {"start": 1098.28, "end": 1099.32, "word": " the", "probability": 0.85400390625}, {"start": 1099.32, "end": 1099.6, "word": " weights", "probability": 0.77490234375}, {"start": 1099.6, "end": 1099.88, "word": " of", "probability": 0.9716796875}, {"start": 1099.88, "end": 1100.5, "word": " catfish", "probability": 0.95166015625}, {"start": 1100.5, "end": 1101.58, "word": " are", "probability": 0.935546875}, {"start": 1101.58, "end": 1102.0, "word": " normally", "probability": 0.90869140625}, {"start": 1102.0, "end": 1102.64, "word": " distributed.", "probability": 0.91552734375}, {"start": 1103.58, "end": 1104.16, "word": " In", "probability": 0.93359375}, {"start": 1104.16, "end": 1104.34, "word": " this", "probability": 0.947265625}, {"start": 1104.34, "end": 1104.58, "word": " case,", "probability": 0.90576171875}, {"start": 1104.66, "end": 1104.7, "word": " you", "probability": 0.338623046875}, {"start": 1104.7, "end": 1104.9, "word": " ask", "probability": 0.91796875}, {"start": 1104.9, "end": 1105.26, "word": " about", "probability": 0.90283203125}, {"start": 1105.26, "end": 1106.7, "word": " what's", "probability": 0.815185546875}, {"start": 1106.7, "end": 1106.84, "word": " the", "probability": 0.88037109375}, {"start": 1106.84, "end": 1107.22, "word": " probability", "probability": 0.98046875}, {"start": 1107.22, "end": 1108.26, "word": " that", "probability": 0.90673828125}, {"start": 1108.26, "end": 1108.6, "word": " a", "probability": 0.890625}, {"start": 1108.6, "end": 1108.98, "word": " randomly", "probability": 0.85302734375}, {"start": 1108.98, "end": 1109.68, "word": " selected", "probability": 0.81787109375}, {"start": 1109.68, "end": 1110.44, "word": " catfish", "probability": 0.966064453125}, {"start": 1110.44, "end": 1111.02, "word": " will", "probability": 0.87353515625}, {"start": 1111.02, "end": 1111.32, "word": " weigh", "probability": 0.845703125}, {"start": 1111.32, "end": 1111.8, "word": " more", "probability": 0.943359375}, {"start": 1111.8, "end": 1112.26, "word": " than", "probability": 0.9501953125}, {"start": 1112.26, "end": 1113.02, "word": " 4", "probability": 0.98486328125}, {"start": 1113.02, "end": 1113.5, "word": ".4.", "probability": 0.99072265625}, {"start": 1113.9, "end": 1114.66, "word": " So", "probability": 0.8916015625}, {"start": 1114.66, "end": 1115.16, "word": " what's", "probability": 0.89306640625}, {"start": 1115.16, "end": 1115.28, "word": " the", "probability": 0.86669921875}, {"start": 1115.28, "end": 1115.62, "word": " probability", "probability": 0.9736328125}, {"start": 1115.62, "end": 1116.68, "word": " of", "probability": 0.9609375}, {"start": 1116.68, "end": 1117.06, "word": " X", "probability": 0.7001953125}], "temperature": 1.0}, {"id": 41, "seek": 114741, "start": 1118.51, "end": 1147.41, "text": " More than. So greater than 4. I just gave the idea to solve this problem. At home, you can compute it to find the exact answer. So first step, find z score. z is 4.4. Divide by z. Just compute this value.", "tokens": [5048, 813, 13, 407, 5044, 813, 1017, 13, 286, 445, 2729, 264, 1558, 281, 5039, 341, 1154, 13, 1711, 1280, 11, 291, 393, 14722, 309, 281, 915, 264, 1900, 1867, 13, 407, 700, 1823, 11, 915, 710, 6175, 13, 710, 307, 1017, 13, 19, 13, 9886, 482, 538, 710, 13, 1449, 14722, 341, 2158, 13], "avg_logprob": -0.20424106690500463, "compression_ratio": 1.3576158940397351, "no_speech_prob": 0.0, "words": [{"start": 1118.51, "end": 1118.87, "word": " More", "probability": 0.261474609375}, {"start": 1118.87, "end": 1119.09, "word": " than.", "probability": 0.923828125}, {"start": 1119.57, "end": 1119.73, "word": " So", "probability": 0.7509765625}, {"start": 1119.73, "end": 1120.07, "word": " greater", "probability": 0.75048828125}, {"start": 1120.07, "end": 1120.77, "word": " than", "probability": 0.92431640625}, {"start": 1120.77, "end": 1121.09, "word": " 4.", "probability": 0.64892578125}, {"start": 1125.83, "end": 1126.39, "word": " I", "probability": 0.81884765625}, {"start": 1126.39, "end": 1126.73, "word": " just", "probability": 0.8798828125}, {"start": 1126.73, "end": 1127.19, "word": " gave", "probability": 0.74853515625}, {"start": 1127.19, "end": 1127.41, "word": " the", "probability": 0.84912109375}, {"start": 1127.41, "end": 1127.79, "word": " idea", "probability": 0.92626953125}, {"start": 1127.79, "end": 1128.01, "word": " to", "probability": 0.82421875}, {"start": 1128.01, "end": 1128.21, "word": " solve", "probability": 0.93359375}, {"start": 1128.21, "end": 1128.39, "word": " this", "probability": 0.80517578125}, {"start": 1128.39, "end": 1128.81, "word": " problem.", "probability": 0.8740234375}, {"start": 1129.09, "end": 1129.29, "word": " At", "probability": 0.77197265625}, {"start": 1129.29, "end": 1129.49, "word": " home,", "probability": 0.8837890625}, {"start": 1129.53, "end": 1129.67, "word": " you", "probability": 0.9619140625}, {"start": 1129.67, "end": 1129.85, "word": " can", "probability": 0.93994140625}, {"start": 1129.85, "end": 1130.13, "word": " compute", "probability": 0.50341796875}, {"start": 1130.13, "end": 1131.13, "word": " it", "probability": 0.73828125}, {"start": 1131.13, "end": 1133.21, "word": " to", "probability": 0.87744140625}, {"start": 1133.21, "end": 1133.43, "word": " find", "probability": 0.896484375}, {"start": 1133.43, "end": 1133.59, "word": " the", "probability": 0.916015625}, {"start": 1133.59, "end": 1133.87, "word": " exact", "probability": 0.927734375}, {"start": 1133.87, "end": 1134.19, "word": " answer.", "probability": 0.951171875}, {"start": 1134.59, "end": 1134.83, "word": " So", "probability": 0.900390625}, {"start": 1134.83, "end": 1135.07, "word": " first", "probability": 0.794921875}, {"start": 1135.07, "end": 1135.33, "word": " step,", "probability": 0.880859375}, {"start": 1135.39, "end": 1135.59, "word": " find", "probability": 0.86181640625}, {"start": 1135.59, "end": 1135.75, "word": " z", "probability": 0.703125}, {"start": 1135.75, "end": 1136.07, "word": " score.", "probability": 0.466064453125}, {"start": 1137.35, "end": 1137.85, "word": " z", "probability": 0.68408203125}, {"start": 1137.85, "end": 1138.15, "word": " is", "probability": 0.94482421875}, {"start": 1138.15, "end": 1138.59, "word": " 4", "probability": 0.99169921875}, {"start": 1138.59, "end": 1139.57, "word": ".4.", "probability": 0.989990234375}, {"start": 1141.73, "end": 1142.29, "word": " Divide", "probability": 0.8466796875}, {"start": 1142.29, "end": 1142.49, "word": " by", "probability": 0.9560546875}, {"start": 1142.49, "end": 1142.59, "word": " z.", "probability": 0.68896484375}, {"start": 1145.91, "end": 1146.47, "word": " Just", "probability": 0.85009765625}, {"start": 1146.47, "end": 1146.81, "word": " compute", "probability": 0.9482421875}, {"start": 1146.81, "end": 1147.07, "word": " this", "probability": 0.951171875}, {"start": 1147.07, "end": 1147.41, "word": " value.", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 42, "seek": 117776, "start": 1148.68, "end": 1177.76, "text": " It's 0.8 divided by 0.8 equals 1. So z-score is 1. So we are looking for the probability of z greater than 1. 1.5. 1.2. 1.5. 1.5. 1.5. So I'm looking for the probability of x of z greater than", "tokens": [467, 311, 1958, 13, 23, 6666, 538, 1958, 13, 23, 6915, 502, 13, 407, 710, 12, 4417, 418, 307, 502, 13, 407, 321, 366, 1237, 337, 264, 8482, 295, 710, 5044, 813, 502, 13, 502, 13, 20, 13, 502, 13, 17, 13, 502, 13, 20, 13, 502, 13, 20, 13, 502, 13, 20, 13, 407, 286, 478, 1237, 337, 264, 8482, 295, 2031, 295, 710, 5044, 813], "avg_logprob": -0.2401194923064288, "compression_ratio": 1.5950413223140496, "no_speech_prob": 0.0, "words": [{"start": 1148.68, "end": 1149.08, "word": " It's", "probability": 0.7470703125}, {"start": 1149.08, "end": 1149.34, "word": " 0", "probability": 0.463623046875}, {"start": 1149.34, "end": 1149.72, "word": ".8", "probability": 0.99267578125}, {"start": 1149.72, "end": 1151.54, "word": " divided", "probability": 0.46875}, {"start": 1151.54, "end": 1151.78, "word": " by", "probability": 0.9677734375}, {"start": 1151.78, "end": 1152.06, "word": " 0", "probability": 0.97412109375}, {"start": 1152.06, "end": 1152.36, "word": ".8", "probability": 0.99853515625}, {"start": 1152.36, "end": 1152.8, "word": " equals", "probability": 0.87890625}, {"start": 1152.8, "end": 1153.1, "word": " 1.", "probability": 0.71826171875}, {"start": 1158.56, "end": 1159.16, "word": " So", "probability": 0.93408203125}, {"start": 1159.16, "end": 1159.4, "word": " z", "probability": 0.390625}, {"start": 1159.4, "end": 1159.62, "word": "-score", "probability": 0.6454264322916666}, {"start": 1159.62, "end": 1159.8, "word": " is", "probability": 0.9501953125}, {"start": 1159.8, "end": 1160.08, "word": " 1.", "probability": 0.82568359375}, {"start": 1161.22, "end": 1161.66, "word": " So", "probability": 0.95556640625}, {"start": 1161.66, "end": 1161.82, "word": " we", "probability": 0.87255859375}, {"start": 1161.82, "end": 1161.94, "word": " are", "probability": 0.9228515625}, {"start": 1161.94, "end": 1162.24, "word": " looking", "probability": 0.92236328125}, {"start": 1162.24, "end": 1162.48, "word": " for", "probability": 0.94775390625}, {"start": 1162.48, "end": 1162.66, "word": " the", "probability": 0.8994140625}, {"start": 1162.66, "end": 1163.0, "word": " probability", "probability": 0.66357421875}, {"start": 1163.0, "end": 1163.24, "word": " of", "probability": 0.90185546875}, {"start": 1163.24, "end": 1163.38, "word": " z", "probability": 0.958984375}, {"start": 1163.38, "end": 1163.8, "word": " greater", "probability": 0.91015625}, {"start": 1163.8, "end": 1164.08, "word": " than", "probability": 0.931640625}, {"start": 1164.08, "end": 1164.7, "word": " 1.", "probability": 0.79638671875}, {"start": 1168.34, "end": 1168.34, "word": " 1", "probability": 0.12213134765625}, {"start": 1168.34, "end": 1168.9, "word": ".5.", "probability": 0.70751953125}, {"start": 1170.02, "end": 1170.62, "word": " 1", "probability": 0.79541015625}, {"start": 1170.62, "end": 1171.16, "word": ".2.", "probability": 0.86669921875}, {"start": 1171.26, "end": 1171.6, "word": " 1", "probability": 0.9287109375}, {"start": 1171.6, "end": 1171.98, "word": ".5.", "probability": 0.930908203125}, {"start": 1172.52, "end": 1172.64, "word": " 1", "probability": 0.8095703125}, {"start": 1172.64, "end": 1173.08, "word": ".5.", "probability": 0.996337890625}, {"start": 1173.18, "end": 1173.48, "word": " 1", "probability": 0.55419921875}, {"start": 1173.48, "end": 1174.02, "word": ".5.", "probability": 0.99853515625}, {"start": 1174.18, "end": 1174.44, "word": " So", "probability": 0.79736328125}, {"start": 1174.44, "end": 1174.58, "word": " I'm", "probability": 0.6217041015625}, {"start": 1174.58, "end": 1174.8, "word": " looking", "probability": 0.91552734375}, {"start": 1174.8, "end": 1175.06, "word": " for", "probability": 0.951171875}, {"start": 1175.06, "end": 1175.22, "word": " the", "probability": 0.89697265625}, {"start": 1175.22, "end": 1175.56, "word": " probability", "probability": 0.95751953125}, {"start": 1175.56, "end": 1175.82, "word": " of", "probability": 0.87939453125}, {"start": 1175.82, "end": 1176.08, "word": " x", "probability": 0.61181640625}, {"start": 1176.08, "end": 1176.32, "word": " of", "probability": 0.9013671875}, {"start": 1176.32, "end": 1176.58, "word": " z", "probability": 0.9912109375}, {"start": 1176.58, "end": 1177.28, "word": " greater", "probability": 0.90234375}, {"start": 1177.28, "end": 1177.76, "word": " than", "probability": 0.94775390625}], "temperature": 1.0}, {"id": 43, "seek": 120726, "start": 1180.98, "end": 1207.26, "text": " 1 minus P of Z less than or equal to 1.5. Now go back to the table. Now 1.5 under 0. It's 0.9332.", "tokens": [502, 3175, 430, 295, 1176, 1570, 813, 420, 2681, 281, 502, 13, 20, 13, 823, 352, 646, 281, 264, 3199, 13, 823, 502, 13, 20, 833, 1958, 13, 467, 311, 1958, 13, 24, 10191, 17, 13], "avg_logprob": -0.2445101448007532, "compression_ratio": 1.0425531914893618, "no_speech_prob": 0.0, "words": [{"start": 1180.98, "end": 1181.66, "word": " 1", "probability": 0.3203125}, {"start": 1181.66, "end": 1182.18, "word": " minus", "probability": 0.367919921875}, {"start": 1182.18, "end": 1188.7, "word": " P", "probability": 0.2435302734375}, {"start": 1188.7, "end": 1188.88, "word": " of", "probability": 0.87890625}, {"start": 1188.88, "end": 1189.2, "word": " Z", "probability": 0.87841796875}, {"start": 1189.2, "end": 1190.2, "word": " less", "probability": 0.76416015625}, {"start": 1190.2, "end": 1190.34, "word": " than", "probability": 0.91259765625}, {"start": 1190.34, "end": 1190.56, "word": " or", "probability": 0.5712890625}, {"start": 1190.56, "end": 1190.62, "word": " equal", "probability": 0.89404296875}, {"start": 1190.62, "end": 1191.42, "word": " to", "probability": 0.9365234375}, {"start": 1191.42, "end": 1191.42, "word": " 1", "probability": 0.59912109375}, {"start": 1191.42, "end": 1191.8, "word": ".5.", "probability": 0.751220703125}, {"start": 1191.8, "end": 1192.04, "word": " Now", "probability": 0.6494140625}, {"start": 1192.04, "end": 1192.24, "word": " go", "probability": 0.689453125}, {"start": 1192.24, "end": 1192.46, "word": " back", "probability": 0.8779296875}, {"start": 1192.46, "end": 1192.58, "word": " to", "probability": 0.96630859375}, {"start": 1192.58, "end": 1192.7, "word": " the", "probability": 0.92578125}, {"start": 1192.7, "end": 1193.04, "word": " table.", "probability": 0.89697265625}, {"start": 1201.54, "end": 1202.22, "word": " Now", "probability": 0.92138671875}, {"start": 1202.22, "end": 1202.44, "word": " 1", "probability": 0.7841796875}, {"start": 1202.44, "end": 1202.98, "word": ".5", "probability": 0.9951171875}, {"start": 1202.98, "end": 1203.28, "word": " under", "probability": 0.90087890625}, {"start": 1203.28, "end": 1203.68, "word": " 0.", "probability": 0.6005859375}, {"start": 1205.0, "end": 1205.68, "word": " It's", "probability": 0.923583984375}, {"start": 1205.68, "end": 1205.9, "word": " 0", "probability": 0.83837890625}, {"start": 1205.9, "end": 1207.26, "word": ".9332.", "probability": 0.960205078125}], "temperature": 1.0}, {"id": 44, "seek": 123755, "start": 1211.41, "end": 1237.55, "text": " So, 1 minus this probability gives 0.668. That's the probability of X greater than 4.4. So, the answer is 0.0668. Now, for the same question.", "tokens": [407, 11, 502, 3175, 341, 8482, 2709, 1958, 13, 15237, 23, 13, 663, 311, 264, 8482, 295, 1783, 5044, 813, 1017, 13, 19, 13, 407, 11, 264, 1867, 307, 1958, 13, 12791, 27102, 13, 823, 11, 337, 264, 912, 1168, 13], "avg_logprob": -0.2157738073950722, "compression_ratio": 1.2456140350877194, "no_speech_prob": 0.0, "words": [{"start": 1211.4099999999999, "end": 1212.09, "word": " So,", "probability": 0.326416015625}, {"start": 1212.25, "end": 1212.29, "word": " 1", "probability": 0.609375}, {"start": 1212.29, "end": 1212.79, "word": " minus", "probability": 0.89990234375}, {"start": 1212.79, "end": 1215.73, "word": " this", "probability": 0.7822265625}, {"start": 1215.73, "end": 1216.23, "word": " probability", "probability": 0.93603515625}, {"start": 1216.23, "end": 1216.81, "word": " gives", "probability": 0.81640625}, {"start": 1216.81, "end": 1218.15, "word": " 0", "probability": 0.94091796875}, {"start": 1218.15, "end": 1219.75, "word": ".668.", "probability": 0.712646484375}, {"start": 1221.21, "end": 1221.89, "word": " That's", "probability": 0.8515625}, {"start": 1221.89, "end": 1222.03, "word": " the", "probability": 0.8759765625}, {"start": 1222.03, "end": 1222.25, "word": " probability", "probability": 0.94189453125}, {"start": 1222.25, "end": 1222.47, "word": " of", "probability": 0.93212890625}, {"start": 1222.47, "end": 1222.65, "word": " X", "probability": 0.736328125}, {"start": 1222.65, "end": 1222.99, "word": " greater", "probability": 0.88720703125}, {"start": 1222.99, "end": 1223.41, "word": " than", "probability": 0.94580078125}, {"start": 1223.41, "end": 1223.75, "word": " 4", "probability": 0.96728515625}, {"start": 1223.75, "end": 1224.35, "word": ".4.", "probability": 0.988037109375}, {"start": 1228.13, "end": 1228.81, "word": " So,", "probability": 0.931640625}, {"start": 1228.87, "end": 1228.93, "word": " the", "probability": 0.91845703125}, {"start": 1228.93, "end": 1229.17, "word": " answer", "probability": 0.9580078125}, {"start": 1229.17, "end": 1229.41, "word": " is", "probability": 0.94140625}, {"start": 1229.41, "end": 1229.59, "word": " 0", "probability": 0.92041015625}, {"start": 1229.59, "end": 1231.59, "word": ".0668.", "probability": 0.8068033854166666}, {"start": 1234.87, "end": 1235.55, "word": " Now,", "probability": 0.916015625}, {"start": 1235.67, "end": 1235.79, "word": " for", "probability": 0.939453125}, {"start": 1235.79, "end": 1235.95, "word": " the", "probability": 0.92236328125}, {"start": 1235.95, "end": 1236.17, "word": " same", "probability": 0.900390625}, {"start": 1236.17, "end": 1237.55, "word": " question.", "probability": 0.65966796875}], "temperature": 1.0}, {"id": 45, "seek": 126832, "start": 1241.32, "end": 1268.32, "text": " What's the probability that a randomly selected fish will weigh between 3 and 5 pounds? 3? Up to 5. So first we have to find the score for 3 out of 5. 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Negative 0.2 divided by 0.8 minus 0.25. The other one, 5 minus 3.2 divided by 0.8. 1 minus 0.8 divided by 0.8 equals", "tokens": [9886, 482, 538, 1958, 13, 23, 307, 264, 700, 710, 2158, 13, 43230, 1958, 13, 17, 6666, 538, 1958, 13, 23, 3175, 1958, 13, 6074, 13, 440, 661, 472, 11, 1025, 3175, 805, 13, 17, 6666, 538, 1958, 13, 23, 13, 502, 3175, 1958, 13, 23, 6666, 538, 1958, 13, 23, 6915], "avg_logprob": -0.19103773359982473, "compression_ratio": 1.4711538461538463, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1270.72, "end": 1271.1, "word": " Divide", "probability": 0.6146240234375}, {"start": 1271.1, "end": 1271.26, "word": " by", "probability": 0.9443359375}, {"start": 1271.26, "end": 1271.52, "word": " 0", "probability": 0.66259765625}, {"start": 1271.52, "end": 1271.82, "word": ".8", "probability": 0.98681640625}, {"start": 1271.82, "end": 1272.02, "word": " is", "probability": 0.391357421875}, {"start": 1272.02, "end": 1272.2, "word": " the", "probability": 0.89990234375}, {"start": 1272.2, "end": 1272.5, "word": " first", "probability": 0.85546875}, {"start": 1272.5, "end": 1272.74, "word": " z", "probability": 0.67236328125}, {"start": 1272.74, "end": 1273.1, "word": " value.", "probability": 0.8486328125}, {"start": 1275.14, "end": 1275.78, "word": " Negative", "probability": 0.74853515625}, {"start": 1275.78, "end": 1276.1, "word": " 0", "probability": 0.95947265625}, {"start": 1276.1, "end": 1276.48, "word": ".2", "probability": 0.99462890625}, {"start": 1276.48, "end": 1277.16, "word": " divided", "probability": 0.480224609375}, {"start": 1277.16, "end": 1277.38, "word": " by", "probability": 0.97607421875}, {"start": 1277.38, "end": 1277.66, "word": " 0", "probability": 0.97705078125}, {"start": 1277.66, "end": 1278.08, "word": ".8", "probability": 0.997802734375}, {"start": 1278.08, "end": 1280.0, "word": " minus", "probability": 0.81005859375}, {"start": 1280.0, "end": 1282.08, "word": " 0", "probability": 0.8447265625}, {"start": 1282.08, "end": 1283.1, "word": ".25.", "probability": 0.964599609375}, {"start": 1284.6, "end": 1285.24, "word": " The", "probability": 0.890625}, {"start": 1285.24, "end": 1285.48, "word": " other", "probability": 0.89013671875}, {"start": 1285.48, "end": 1285.86, "word": " one,", "probability": 0.92138671875}, {"start": 1288.44, "end": 1289.2, "word": " 5", "probability": 0.81689453125}, {"start": 1289.2, "end": 1289.46, "word": " minus", "probability": 0.9384765625}, {"start": 1289.46, "end": 1289.66, "word": " 3", "probability": 0.9228515625}, {"start": 1289.66, "end": 1290.08, "word": ".2", "probability": 0.982421875}, {"start": 1290.08, "end": 1290.36, "word": " divided", "probability": 0.791015625}, {"start": 1290.36, "end": 1290.54, "word": " by", "probability": 0.97607421875}, {"start": 1290.54, "end": 1290.8, "word": " 0", "probability": 0.92822265625}, {"start": 1290.8, "end": 1291.16, "word": ".8.", "probability": 0.99853515625}, {"start": 1292.68, "end": 1293.32, "word": " 1", "probability": 0.91552734375}, {"start": 1293.32, "end": 1293.64, "word": " minus", "probability": 0.96435546875}, {"start": 1293.64, "end": 1293.88, "word": " 0", "probability": 0.97607421875}, {"start": 1293.88, "end": 1294.12, "word": ".8", "probability": 0.998046875}, {"start": 1294.12, "end": 1294.38, "word": " divided", "probability": 0.85546875}, {"start": 1294.38, "end": 1294.62, "word": " by", "probability": 0.9736328125}, {"start": 1294.62, "end": 1294.88, "word": " 0", "probability": 0.9736328125}, {"start": 1294.88, "end": 1295.32, "word": ".8", "probability": 0.99951171875}, {"start": 1295.32, "end": 1296.68, "word": " equals", "probability": 0.88525390625}], "temperature": 1.0}, {"id": 47, "seek": 132820, "start": 1302.68, "end": 1328.2, "text": " 2.25. Just double check this result. So here, the probability of X between 3 and 5 equals minus 0.25, smaller than Z, smaller than 2.5.", "tokens": [568, 13, 6074, 13, 1449, 3834, 1520, 341, 1874, 13, 407, 510, 11, 264, 8482, 295, 1783, 1296, 805, 293, 1025, 6915, 3175, 1958, 13, 6074, 11, 4356, 813, 1176, 11, 4356, 813, 568, 13, 20, 13], "avg_logprob": -0.2993421154586892, "compression_ratio": 1.2142857142857142, "no_speech_prob": 0.0, "words": [{"start": 1302.6799999999998, "end": 1303.3999999999999, "word": " 2", "probability": 0.1766357421875}, {"start": 1303.3999999999999, "end": 1304.12, "word": ".25.", "probability": 0.924560546875}, {"start": 1310.84, "end": 1311.56, "word": " Just", "probability": 0.6767578125}, {"start": 1311.56, "end": 1311.76, "word": " double", "probability": 0.58935546875}, {"start": 1311.76, "end": 1312.02, "word": " check", "probability": 0.6484375}, {"start": 1312.02, "end": 1312.28, "word": " this", "probability": 0.9208984375}, {"start": 1312.28, "end": 1312.66, "word": " result.", "probability": 0.95166015625}, {"start": 1314.9, "end": 1315.62, "word": " So", "probability": 0.6494140625}, {"start": 1315.62, "end": 1316.22, "word": " here,", "probability": 0.6279296875}, {"start": 1316.6, "end": 1317.02, "word": " the", "probability": 0.912109375}, {"start": 1317.02, "end": 1318.66, "word": " probability", "probability": 0.2607421875}, {"start": 1318.66, "end": 1318.82, "word": " of", "probability": 0.94189453125}, {"start": 1318.82, "end": 1319.0, "word": " X", "probability": 0.84423828125}, {"start": 1319.0, "end": 1319.3, "word": " between", "probability": 0.89013671875}, {"start": 1319.3, "end": 1319.56, "word": " 3", "probability": 0.71923828125}, {"start": 1319.56, "end": 1319.66, "word": " and", "probability": 0.94384765625}, {"start": 1319.66, "end": 1320.06, "word": " 5", "probability": 0.98779296875}, {"start": 1320.06, "end": 1321.76, "word": " equals", "probability": 0.8466796875}, {"start": 1321.76, "end": 1324.24, "word": " minus", "probability": 0.76513671875}, {"start": 1324.24, "end": 1324.52, "word": " 0", "probability": 0.67138671875}, {"start": 1324.52, "end": 1325.14, "word": ".25,", "probability": 0.98779296875}, {"start": 1325.74, "end": 1326.44, "word": " smaller", "probability": 0.83203125}, {"start": 1326.44, "end": 1326.74, "word": " than", "probability": 0.9462890625}, {"start": 1326.74, "end": 1327.0, "word": " Z,", "probability": 0.830078125}, {"start": 1327.2, "end": 1327.5, "word": " smaller", "probability": 0.7861328125}, {"start": 1327.5, "end": 1327.7, "word": " than", "probability": 0.94970703125}, {"start": 1327.7, "end": 1327.86, "word": " 2", "probability": 0.94580078125}, {"start": 1327.86, "end": 1328.2, "word": ".5.", "probability": 0.70068359375}], "temperature": 1.0}, {"id": 48, "seek": 135809, "start": 1332.65, "end": 1358.09, "text": " So it's transformed from normal distribution to standardized normal distribution. So here instead of computing the probability of X between three and five, we are looking for the probability between Z between actually minus. It's minus because your value here is smaller than the average. The average was 3.2, so it should be negative.", "tokens": [407, 309, 311, 16894, 490, 2710, 7316, 281, 31677, 2710, 7316, 13, 407, 510, 2602, 295, 15866, 264, 8482, 295, 1783, 1296, 1045, 293, 1732, 11, 321, 366, 1237, 337, 264, 8482, 1296, 1176, 1296, 767, 3175, 13, 467, 311, 3175, 570, 428, 2158, 510, 307, 4356, 813, 264, 4274, 13, 440, 4274, 390, 805, 13, 17, 11, 370, 309, 820, 312, 3671, 13], "avg_logprob": -0.1753605769230769, "compression_ratio": 1.6231884057971016, "no_speech_prob": 0.0, "words": [{"start": 1332.65, "end": 1333.13, "word": " So", "probability": 0.8056640625}, {"start": 1333.13, "end": 1333.79, "word": " it's", "probability": 0.791748046875}, {"start": 1333.79, "end": 1334.55, "word": " transformed", "probability": 0.87353515625}, {"start": 1334.55, "end": 1335.25, "word": " from", "probability": 0.88232421875}, {"start": 1335.25, "end": 1336.09, "word": " normal", "probability": 0.69677734375}, {"start": 1336.09, "end": 1336.77, "word": " distribution", 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"probability": 0.9501953125}, {"start": 1357.79, "end": 1358.09, "word": " negative.", "probability": 0.896484375}], "temperature": 1.0}, {"id": 49, "seek": 138587, "start": 1359.63, "end": 1385.87, "text": " So z score between minus 0.25 all the way up to 2.25. So now, this is the probability we are looking for. Zero in the middle minus one-fourth to the left of z of zero, mu of zero. And this is the value of 2.25.", "tokens": [407, 710, 6175, 1296, 3175, 1958, 13, 6074, 439, 264, 636, 493, 281, 568, 13, 6074, 13, 407, 586, 11, 341, 307, 264, 8482, 321, 366, 1237, 337, 13, 17182, 294, 264, 2808, 3175, 472, 12, 23251, 392, 281, 264, 1411, 295, 710, 295, 4018, 11, 2992, 295, 4018, 13, 400, 341, 307, 264, 2158, 295, 568, 13, 6074, 13], "avg_logprob": -0.24641392660922692, "compression_ratio": 1.4256756756756757, "no_speech_prob": 0.0, "words": [{"start": 1359.63, "end": 1360.01, "word": " So", "probability": 0.5732421875}, {"start": 1360.01, "end": 1360.25, "word": " z", "probability": 0.1942138671875}, {"start": 1360.25, "end": 1360.49, "word": " score", "probability": 0.2239990234375}, {"start": 1360.49, "end": 1360.79, "word": " between", "probability": 0.8076171875}, {"start": 1360.79, "end": 1361.13, "word": " minus", "probability": 0.48486328125}, {"start": 1361.13, "end": 1361.35, "word": " 0", "probability": 0.27490234375}, {"start": 1361.35, "end": 1361.91, "word": ".25", "probability": 0.985107421875}, {"start": 1361.91, "end": 1362.47, "word": " all", "probability": 0.70556640625}, {"start": 1362.47, "end": 1362.63, "word": " the", "probability": 0.892578125}, {"start": 1362.63, "end": 1362.79, "word": " way", "probability": 0.95166015625}, {"start": 1362.79, "end": 1362.97, "word": " up", "probability": 0.85546875}, {"start": 1362.97, "end": 1363.11, "word": " to", "probability": 0.94677734375}, {"start": 1363.11, "end": 1363.29, "word": " 2", "probability": 0.93994140625}, {"start": 1363.29, "end": 1363.75, "word": ".25.", "probability": 0.956298828125}, {"start": 1365.07, "end": 1365.63, "word": " So", "probability": 0.91357421875}, {"start": 1365.63, "end": 1365.91, "word": " now,", "probability": 0.84912109375}, {"start": 1366.77, "end": 1367.23, "word": " this", "probability": 0.9267578125}, {"start": 1367.23, "end": 1367.35, "word": " is", "probability": 0.94189453125}, {"start": 1367.35, "end": 1367.49, "word": " the", "probability": 0.84521484375}, {"start": 1367.49, "end": 1367.89, "word": " probability", "probability": 0.9462890625}, {"start": 1367.89, "end": 1368.15, "word": " we", "probability": 0.890625}, {"start": 1368.15, "end": 1368.27, "word": " are", "probability": 0.9033203125}, {"start": 1368.27, "end": 1368.53, "word": " looking", "probability": 0.916015625}, {"start": 1368.53, "end": 1368.89, "word": " for.", "probability": 0.95703125}, {"start": 1370.85, "end": 1371.41, "word": " Zero", "probability": 0.6025390625}, {"start": 1371.41, "end": 1372.09, "word": " in", "probability": 0.865234375}, {"start": 1372.09, "end": 1372.21, "word": " the", "probability": 0.91748046875}, {"start": 1372.21, "end": 1372.55, "word": " middle", "probability": 0.94482421875}, {"start": 1372.55, "end": 1373.63, "word": " minus", "probability": 0.75341796875}, {"start": 1373.63, "end": 1376.59, "word": " one", "probability": 0.5947265625}, {"start": 1376.59, "end": 1378.37, "word": "-fourth", "probability": 0.8058268229166666}, {"start": 1378.37, "end": 1378.65, "word": " to", "probability": 0.72265625}, {"start": 1378.65, "end": 1378.75, "word": " the", "probability": 0.91796875}, {"start": 1378.75, "end": 1378.97, "word": " left", "probability": 0.93505859375}, {"start": 1378.97, "end": 1379.17, "word": " of", "probability": 0.96142578125}, {"start": 1379.17, "end": 1379.39, "word": " z", "probability": 0.8232421875}, {"start": 1379.39, "end": 1380.17, "word": " of", "probability": 0.7197265625}, {"start": 1380.17, "end": 1380.53, "word": " zero,", "probability": 0.81396484375}, {"start": 1381.51, "end": 1381.79, "word": " mu", "probability": 0.53125}, {"start": 1381.79, "end": 1381.95, "word": " of", "probability": 0.86474609375}, {"start": 1381.95, "end": 1382.19, "word": " zero.", "probability": 0.91015625}, {"start": 1383.23, "end": 1383.61, "word": " And", "probability": 0.9130859375}, {"start": 1383.61, "end": 1383.79, "word": " this", "probability": 0.82421875}, {"start": 1383.79, "end": 1383.91, "word": " is", "probability": 0.921875}, {"start": 1383.91, "end": 1384.01, "word": " the", "probability": 0.86669921875}, {"start": 1384.01, "end": 1384.27, "word": " value", "probability": 0.97509765625}, {"start": 1384.27, "end": 1384.49, "word": " of", "probability": 0.96435546875}, {"start": 1384.49, "end": 1384.75, "word": " 2", "probability": 0.9609375}, {"start": 1384.75, "end": 1385.87, "word": ".25.", "probability": 0.98876953125}], "temperature": 1.0}, {"id": 50, "seek": 141668, "start": 1387.7, "end": 1416.68, "text": " Now we are looking actually for this probability. The area between minus 0.25 all the way up to 2.5. So this area equals the probability of Z less than 2.25 minus. And again, use the normal.", "tokens": [823, 321, 366, 1237, 767, 337, 341, 8482, 13, 440, 1859, 1296, 3175, 1958, 13, 6074, 439, 264, 636, 493, 281, 568, 13, 20, 13, 407, 341, 1859, 6915, 264, 8482, 295, 1176, 1570, 813, 568, 13, 6074, 3175, 13, 400, 797, 11, 764, 264, 2710, 13], "avg_logprob": -0.18554687841484943, "compression_ratio": 1.3642857142857143, "no_speech_prob": 0.0, "words": [{"start": 1387.7, "end": 1388.0, "word": " Now", "probability": 0.88818359375}, {"start": 1388.0, "end": 1388.18, "word": " we", "probability": 0.6962890625}, {"start": 1388.18, "end": 1388.3, "word": " are", "probability": 0.91796875}, {"start": 1388.3, "end": 1388.56, "word": " looking", "probability": 0.8759765625}, {"start": 1388.56, "end": 1389.06, "word": " actually", "probability": 0.7666015625}, {"start": 1389.06, "end": 1389.32, "word": " for", "probability": 0.9375}, {"start": 1389.32, "end": 1389.52, "word": " this", "probability": 0.9150390625}, {"start": 1389.52, "end": 1389.94, "word": " probability.", "probability": 0.92431640625}, {"start": 1392.96, "end": 1393.6, "word": " The", "probability": 0.87353515625}, {"start": 1393.6, "end": 1393.86, "word": " area", "probability": 0.90625}, {"start": 1393.86, "end": 1394.36, "word": " between", "probability": 0.8701171875}, {"start": 1394.36, "end": 1395.32, "word": " minus", "probability": 0.87744140625}, {"start": 1395.32, "end": 1395.6, "word": " 0", "probability": 0.81640625}, {"start": 1395.6, "end": 1396.02, "word": ".25", "probability": 0.994140625}, {"start": 1396.02, "end": 1397.06, "word": " all", "probability": 0.85498046875}, {"start": 1397.06, "end": 1397.24, "word": " the", "probability": 0.92041015625}, {"start": 1397.24, "end": 1397.38, "word": " way", "probability": 0.953125}, {"start": 1397.38, "end": 1397.58, "word": " up", "probability": 0.94921875}, {"start": 1397.58, "end": 1397.72, "word": " to", "probability": 0.9638671875}, {"start": 1397.72, "end": 1397.94, "word": " 2", "probability": 0.986328125}, {"start": 1397.94, "end": 1398.36, "word": ".5.", "probability": 0.8134765625}, {"start": 1399.98, "end": 1400.62, "word": " So", "probability": 0.939453125}, {"start": 1400.62, "end": 1400.82, "word": " this", "probability": 0.859375}, {"start": 1400.82, "end": 1401.12, "word": " area", "probability": 0.9169921875}, {"start": 1401.12, "end": 1401.68, "word": " equals", "probability": 0.93994140625}, {"start": 1401.68, "end": 1405.2, "word": " the", "probability": 0.34326171875}, {"start": 1405.2, "end": 1405.48, "word": " probability", "probability": 0.92431640625}, {"start": 1405.48, "end": 1405.8, "word": " of", "probability": 0.953125}, {"start": 1405.8, "end": 1406.02, "word": " Z", "probability": 0.6142578125}, {"start": 1406.02, "end": 1406.68, "word": " less", "probability": 0.84716796875}, {"start": 1406.68, "end": 1407.0, "word": " than", "probability": 0.93994140625}, {"start": 1407.0, "end": 1407.64, "word": " 2", "probability": 0.974609375}, {"start": 1407.64, "end": 1408.36, "word": ".25", "probability": 0.913330078125}, {"start": 1408.36, "end": 1409.0, "word": " minus.", "probability": 0.974609375}, {"start": 1414.28, "end": 1414.92, "word": " And", "probability": 0.262939453125}, {"start": 1414.92, "end": 1415.28, "word": " again,", "probability": 0.93896484375}, {"start": 1415.94, "end": 1416.24, "word": " use", "probability": 0.87109375}, {"start": 1416.24, "end": 1416.4, "word": " the", "probability": 0.91552734375}, {"start": 1416.4, "end": 1416.68, "word": " normal.", "probability": 0.783203125}], "temperature": 1.0}, {"id": 51, "seek": 144278, "start": 1417.08, "end": 1442.78, "text": " table to give this value and another one. Any questions? So first step here, we compute the z-score for each value x. So the problem is transformed from normal distribution to standardized normal distribution.", "tokens": [3199, 281, 976, 341, 2158, 293, 1071, 472, 13, 2639, 1651, 30, 407, 700, 1823, 510, 11, 321, 14722, 264, 710, 12, 4417, 418, 337, 1184, 2158, 2031, 13, 407, 264, 1154, 307, 16894, 490, 2710, 7316, 281, 31677, 2710, 7316, 13], "avg_logprob": -0.23110465740048608, "compression_ratio": 1.4383561643835616, "no_speech_prob": 0.0, "words": [{"start": 1417.08, "end": 1417.5, "word": " table", "probability": 0.3759765625}, {"start": 1417.5, "end": 1418.34, "word": " to", "probability": 0.89501953125}, {"start": 1418.34, "end": 1418.54, "word": " give", "probability": 0.75634765625}, {"start": 1418.54, "end": 1418.78, "word": " this", "probability": 0.935546875}, {"start": 1418.78, "end": 1419.14, "word": " value", "probability": 0.9677734375}, {"start": 1419.14, "end": 1419.38, "word": " and", "probability": 0.8779296875}, {"start": 1419.38, "end": 1419.6, "word": " another", "probability": 0.376220703125}, {"start": 1419.6, "end": 1419.86, "word": " one.", "probability": 0.74560546875}, {"start": 1422.98, "end": 1423.62, "word": " Any", "probability": 0.8349609375}, {"start": 1423.62, "end": 1423.94, "word": " questions?", "probability": 0.496337890625}, {"start": 1425.18, "end": 1425.96, "word": " So", "probability": 0.876953125}, {"start": 1425.96, "end": 1426.26, "word": " first", "probability": 0.66455078125}, {"start": 1426.26, "end": 1426.5, "word": " step", "probability": 0.896484375}, {"start": 1426.5, "end": 1426.78, "word": " here,", "probability": 0.857421875}, {"start": 1427.82, "end": 1428.16, "word": " we", "probability": 0.9501953125}, {"start": 1428.16, "end": 1430.64, "word": " compute", "probability": 0.8916015625}, {"start": 1430.64, "end": 1432.4, "word": " the", "probability": 0.88720703125}, {"start": 1432.4, "end": 1432.68, "word": " z", "probability": 0.7333984375}, {"start": 1432.68, "end": 1433.02, "word": "-score", "probability": 0.7985026041666666}, {"start": 1433.02, "end": 1433.3, "word": " for", "probability": 0.9384765625}, {"start": 1433.3, "end": 1433.52, "word": " each", "probability": 0.93994140625}, {"start": 1433.52, "end": 1433.78, "word": " value", "probability": 0.873046875}, {"start": 1433.78, "end": 1434.14, "word": " x.", "probability": 0.5283203125}, {"start": 1435.6, "end": 1436.1, "word": " So", "probability": 0.95849609375}, {"start": 1436.1, "end": 1436.3, "word": " the", "probability": 0.8525390625}, {"start": 1436.3, "end": 1436.62, "word": " problem", "probability": 0.85205078125}, {"start": 1436.62, "end": 1436.88, "word": " is", "probability": 0.689453125}, {"start": 1436.88, "end": 1437.6, "word": " transformed", "probability": 0.875}, {"start": 1437.6, "end": 1438.02, "word": " from", "probability": 0.88720703125}, {"start": 1438.02, "end": 1438.44, "word": " normal", "probability": 0.8310546875}, {"start": 1438.44, "end": 1439.1, "word": " distribution", "probability": 0.83544921875}, {"start": 1439.1, "end": 1441.38, "word": " to", "probability": 0.6640625}, {"start": 1441.38, "end": 1441.86, "word": " standardized", "probability": 0.80712890625}, {"start": 1441.86, "end": 1442.3, "word": " normal", "probability": 0.84765625}, {"start": 1442.3, "end": 1442.78, "word": " distribution.", "probability": 0.85791015625}], "temperature": 1.0}, {"id": 52, "seek": 147310, "start": 1444.18, "end": 1473.1, "text": " So it becomes z between minus 1.25 up to 2.25. Now, this area, this dashed area equals the area below 2.25 minus the area below minus 1.25. Now, by using the similar way we did before, you will compute the value of z. The probability of z is smaller than 2.25 by using", "tokens": [407, 309, 3643, 710, 1296, 3175, 502, 13, 6074, 493, 281, 568, 13, 6074, 13, 823, 11, 341, 1859, 11, 341, 8240, 292, 1859, 6915, 264, 1859, 2507, 568, 13, 6074, 3175, 264, 1859, 2507, 3175, 502, 13, 6074, 13, 823, 11, 538, 1228, 264, 2531, 636, 321, 630, 949, 11, 291, 486, 14722, 264, 2158, 295, 710, 13, 440, 8482, 295, 710, 307, 4356, 813, 568, 13, 6074, 538, 1228], "avg_logprob": -0.1979166730824444, "compression_ratio": 1.6107784431137724, "no_speech_prob": 0.0, "words": [{"start": 1444.18, "end": 1444.44, "word": " So", "probability": 0.66552734375}, {"start": 1444.44, "end": 1444.56, "word": " it", "probability": 0.75}, {"start": 1444.56, "end": 1444.88, "word": " becomes", "probability": 0.85546875}, {"start": 1444.88, "end": 1445.06, "word": " z", "probability": 0.5732421875}, {"start": 1445.06, "end": 1445.36, "word": " between", "probability": 0.8818359375}, {"start": 1445.36, "end": 1445.74, "word": " minus", 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So here, 2.2 up to 5. So 9, 8, 7, 8. 9, 8, 7, 8. So the area below 2.25, 2.2, this value.", "tokens": [440, 2710, 3199, 13, 407, 510, 11, 568, 13, 17, 493, 281, 1025, 13, 407, 1722, 11, 1649, 11, 1614, 11, 1649, 13, 1722, 11, 1649, 11, 1614, 11, 1649, 13, 407, 264, 1859, 2507, 568, 13, 6074, 11, 568, 13, 17, 11, 341, 2158, 13], "avg_logprob": -0.26213430343790256, "compression_ratio": 1.2, "no_speech_prob": 0.0, "words": [{"start": 1473.84, "end": 1474.16, "word": " The", "probability": 0.1510009765625}, {"start": 1474.16, "end": 1474.44, "word": " normal", "probability": 0.703125}, {"start": 1474.44, "end": 1474.82, "word": " table.", "probability": 0.78369140625}, {"start": 1476.16, "end": 1476.72, "word": " So", "probability": 0.7265625}, {"start": 1476.72, "end": 1476.92, "word": " here,", "probability": 0.642578125}, {"start": 1477.04, "end": 1477.18, "word": " 2", "probability": 0.95556640625}, {"start": 1477.18, "end": 1477.76, "word": ".2", "probability": 0.9755859375}, {"start": 1477.76, "end": 1479.42, "word": " up", "probability": 0.81689453125}, {"start": 1479.42, "end": 1479.58, "word": " to", "probability": 0.94189453125}, {"start": 1479.58, "end": 1479.9, "word": " 5.", "probability": 0.88232421875}, {"start": 1482.58, "end": 1483.14, "word": " So", "probability": 0.79443359375}, {"start": 1483.14, "end": 1483.48, "word": " 9,", "probability": 0.354736328125}, {"start": 1483.76, "end": 1484.26, "word": " 8,", "probability": 0.8603515625}, {"start": 1484.46, "end": 1484.7, "word": " 7,", "probability": 0.98681640625}, {"start": 1484.8, "end": 1485.0, "word": " 8.", "probability": 0.98681640625}, {"start": 1486.42, "end": 1486.98, "word": " 9,", "probability": 0.81640625}, {"start": 1487.04, "end": 1487.26, "word": " 8,", "probability": 0.9873046875}, {"start": 1487.4, "end": 1487.56, "word": " 7,", "probability": 0.998046875}, {"start": 1487.66, "end": 1487.9, "word": " 8.", "probability": 0.99609375}, {"start": 1493.9, "end": 1494.46, "word": " So", "probability": 0.9462890625}, {"start": 1494.46, "end": 1494.68, "word": " the", "probability": 0.75244140625}, {"start": 1494.68, "end": 1494.96, "word": " area", "probability": 0.89892578125}, {"start": 1494.96, "end": 1495.26, "word": " below", "probability": 0.91064453125}, {"start": 1495.26, "end": 1496.52, "word": " 2", "probability": 0.331787109375}, {"start": 1496.52, "end": 1497.6, "word": ".25,", "probability": 0.973876953125}, {"start": 1497.82, "end": 1498.1, "word": " 2", "probability": 0.97265625}, {"start": 1498.1, "end": 1498.5, "word": ".2,", "probability": 0.99365234375}, {"start": 1498.58, "end": 1498.72, "word": " this", "probability": 0.931640625}, {"start": 1498.72, "end": 1499.02, "word": " value.", "probability": 0.9736328125}], "temperature": 1.0}, {"id": 54, "seek": 152902, "start": 1499.76, "end": 1529.02, "text": " All the way up to 5 gives 987. Now, what's about the probability of Z smaller than minus 0.25? If you go back to the Z table, but for the other one, the negative one. Minus 2 minus 0.2 up to 5.", "tokens": [1057, 264, 636, 493, 281, 1025, 2709, 20860, 22, 13, 823, 11, 437, 311, 466, 264, 8482, 295, 1176, 4356, 813, 3175, 1958, 13, 6074, 30, 759, 291, 352, 646, 281, 264, 1176, 3199, 11, 457, 337, 264, 661, 472, 11, 264, 3671, 472, 13, 2829, 301, 568, 3175, 1958, 13, 17, 493, 281, 1025, 13], "avg_logprob": -0.18804824561403508, "compression_ratio": 1.3108108108108107, "no_speech_prob": 0.0, "words": [{"start": 1499.76, "end": 1500.08, "word": " All", "probability": 0.410400390625}, {"start": 1500.08, "end": 1500.26, "word": " the", "probability": 0.91162109375}, {"start": 1500.26, "end": 1500.44, "word": " way", "probability": 0.95703125}, {"start": 1500.44, "end": 1500.66, "word": " up", "probability": 0.92919921875}, {"start": 1500.66, "end": 1500.84, "word": " to", "probability": 0.89892578125}, {"start": 1500.84, "end": 1501.1, "word": " 5", "probability": 0.58447265625}, {"start": 1501.1, "end": 1501.62, "word": " gives", "probability": 0.771484375}, {"start": 1501.62, "end": 1503.94, "word": " 987.", "probability": 0.730712890625}, {"start": 1505.54, "end": 1506.18, "word": " Now,", "probability": 0.93017578125}, {"start": 1506.3, "end": 1506.68, "word": " what's", "probability": 0.861328125}, {"start": 1506.68, "end": 1506.9, "word": " about", "probability": 0.8583984375}, {"start": 1506.9, "end": 1507.1, "word": " the", "probability": 0.8994140625}, {"start": 1507.1, "end": 1507.46, "word": " probability", "probability": 0.95068359375}, {"start": 1507.46, "end": 1507.78, "word": " of", "probability": 0.94921875}, {"start": 1507.78, "end": 1508.08, "word": " Z", "probability": 0.5556640625}, {"start": 1508.08, "end": 1508.86, "word": " smaller", "probability": 0.76123046875}, {"start": 1508.86, "end": 1509.18, "word": " than", "probability": 0.9384765625}, {"start": 1509.18, "end": 1509.56, "word": " minus", "probability": 0.70849609375}, {"start": 1509.56, "end": 1511.22, "word": " 0", "probability": 0.7763671875}, {"start": 1511.22, "end": 1511.82, "word": ".25?", "probability": 0.99462890625}, {"start": 1512.98, "end": 1513.62, "word": " If", "probability": 0.9541015625}, {"start": 1513.62, "end": 1513.74, "word": " you", "probability": 0.95849609375}, {"start": 1513.74, "end": 1513.9, "word": " go", "probability": 0.9580078125}, {"start": 1513.9, "end": 1514.22, "word": " back", "probability": 0.875}, {"start": 1514.22, "end": 1514.62, "word": " to", "probability": 0.96142578125}, {"start": 1514.62, "end": 1514.82, "word": " the", "probability": 0.79248046875}, {"start": 1514.82, "end": 1515.06, "word": " Z", "probability": 0.93310546875}, {"start": 1515.06, "end": 1515.32, "word": " table,", "probability": 0.5302734375}, {"start": 1515.38, "end": 1515.54, "word": " but", "probability": 0.9140625}, {"start": 1515.54, "end": 1515.74, "word": " for", "probability": 0.88623046875}, {"start": 1515.74, "end": 1515.94, "word": " the", "probability": 0.92041015625}, {"start": 1515.94, "end": 1516.16, "word": " other", "probability": 0.87744140625}, {"start": 1516.16, "end": 1516.48, "word": " one,", "probability": 0.92626953125}, {"start": 1518.08, "end": 1518.3, "word": " the", "probability": 0.91015625}, {"start": 1518.3, "end": 1518.62, "word": " negative", "probability": 0.94775390625}, {"start": 1518.62, "end": 1518.96, "word": " one.", "probability": 0.912109375}, {"start": 1523.12, "end": 1523.76, "word": " Minus", "probability": 0.955810546875}, {"start": 1523.76, "end": 1524.28, "word": " 2", "probability": 0.666015625}, {"start": 1524.28, "end": 1524.7, "word": " minus", "probability": 0.4873046875}, {"start": 1524.7, "end": 1525.04, "word": " 0", "probability": 0.95556640625}, {"start": 1525.04, "end": 1525.46, "word": ".2", "probability": 0.982421875}, {"start": 1525.46, "end": 1528.54, "word": " up", "probability": 0.45947265625}, {"start": 1528.54, "end": 1528.7, "word": " to", "probability": 0.9345703125}, {"start": 1528.7, "end": 1529.02, "word": " 5.", "probability": 0.947265625}], "temperature": 1.0}, {"id": 55, "seek": 155246, "start": 1530.26, "end": 1552.46, "text": " 0.4013 minus, that will give the probability between three and five. This is the second part. So the final answer.", "tokens": [1958, 13, 5254, 7668, 3175, 11, 300, 486, 976, 264, 8482, 1296, 1045, 293, 1732, 13, 639, 307, 264, 1150, 644, 13, 407, 264, 2572, 1867, 13], "avg_logprob": -0.34207588114908766, "compression_ratio": 1.1386138613861385, "no_speech_prob": 0.0, "words": [{"start": 1530.26, "end": 1531.3, "word": " 0", "probability": 0.1712646484375}, {"start": 1531.3, "end": 1532.32, "word": ".4013", "probability": 0.9041341145833334}, {"start": 1532.32, "end": 1534.78, "word": " minus,", "probability": 0.810546875}, {"start": 1536.62, "end": 1538.7, "word": " that", "probability": 0.919921875}, {"start": 1538.7, "end": 1538.9, "word": " will", "probability": 0.8896484375}, {"start": 1538.9, "end": 1539.2, "word": " give", "probability": 0.83642578125}, {"start": 1539.2, "end": 1540.0, "word": " the", "probability": 0.8447265625}, {"start": 1540.0, "end": 1540.74, "word": " probability", "probability": 0.9453125}, {"start": 1540.74, "end": 1541.32, "word": " between", "probability": 0.86572265625}, {"start": 1541.32, "end": 1542.92, "word": " three", "probability": 0.180419921875}, {"start": 1542.92, "end": 1543.1, "word": " and", "probability": 0.8046875}, {"start": 1543.1, "end": 1543.28, "word": " five.", "probability": 0.65234375}, {"start": 1546.18, "end": 1547.34, "word": " This", "probability": 0.84765625}, {"start": 1547.34, "end": 1547.48, "word": " is", "probability": 0.94921875}, {"start": 1547.48, "end": 1547.64, "word": " the", "probability": 0.91650390625}, {"start": 1547.64, "end": 1548.06, "word": " second", "probability": 0.900390625}, {"start": 1548.06, "end": 1548.9, "word": " part.", "probability": 0.90625}, {"start": 1551.12, "end": 1551.66, "word": " So", "probability": 0.890625}, {"start": 1551.66, "end": 1551.82, "word": " the", "probability": 0.66357421875}, {"start": 1551.82, "end": 1552.08, "word": " final", "probability": 0.93896484375}, {"start": 1552.08, "end": 1552.46, "word": " answer.", "probability": 0.95703125}], "temperature": 1.0}, {"id": 56, "seek": 158229, "start": 1560.63, "end": 1582.29, "text": " So this is the probability that the selected cash fish will weigh between three and five pounds. Now, other question is, for the same problem, you said a citation", "tokens": [407, 341, 307, 264, 8482, 300, 264, 8209, 6388, 3506, 486, 13843, 1296, 1045, 293, 1732, 8319, 13, 823, 11, 661, 1168, 307, 11, 337, 264, 912, 1154, 11, 291, 848, 257, 45590], "avg_logprob": -0.3446691071285921, "compression_ratio": 1.314516129032258, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 1560.63, "end": 1561.63, "word": " So", "probability": 0.12744140625}, {"start": 1561.63, "end": 1562.63, "word": " this", "probability": 0.783203125}, {"start": 1562.63, "end": 1562.75, "word": " is", "probability": 0.88330078125}, {"start": 1562.75, "end": 1562.83, "word": " the", "probability": 0.47509765625}, {"start": 1562.83, "end": 1563.15, "word": " probability", "probability": 0.95263671875}, {"start": 1563.15, "end": 1563.85, "word": " that", "probability": 0.9091796875}, {"start": 1563.85, "end": 1564.47, "word": " the", "probability": 0.86279296875}, {"start": 1564.47, "end": 1565.09, "word": " selected", "probability": 0.88720703125}, {"start": 1565.09, "end": 1565.45, "word": " cash", "probability": 0.454345703125}, {"start": 1565.45, "end": 1565.87, "word": " fish", "probability": 0.68798828125}, {"start": 1565.87, "end": 1566.61, "word": " will", "probability": 0.86474609375}, {"start": 1566.61, "end": 1566.85, "word": " weigh", "probability": 0.89306640625}, {"start": 1566.85, "end": 1567.35, "word": " between", "probability": 0.8369140625}, {"start": 1567.35, "end": 1568.71, "word": " three", "probability": 0.5458984375}, {"start": 1568.71, "end": 1568.95, "word": " and", "probability": 0.93017578125}, {"start": 1568.95, "end": 1569.49, "word": " five", "probability": 0.90673828125}, {"start": 1569.49, "end": 1570.65, "word": " pounds.", "probability": 0.83056640625}, {"start": 1571.81, "end": 1572.11, "word": " Now,", "probability": 0.82080078125}, {"start": 1573.15, "end": 1573.45, "word": " other", "probability": 0.37109375}, {"start": 1573.45, "end": 1573.83, "word": " question", "probability": 0.89208984375}, {"start": 1573.83, "end": 1574.11, "word": " is,", "probability": 0.92626953125}, {"start": 1574.97, "end": 1577.27, "word": " for", "probability": 0.9189453125}, {"start": 1577.27, "end": 1577.45, "word": " the", "probability": 0.91796875}, {"start": 1577.45, "end": 1577.65, "word": " same", "probability": 0.89404296875}, {"start": 1577.65, "end": 1578.11, "word": " problem,", "probability": 0.861328125}, {"start": 1580.53, "end": 1580.77, "word": " you", "probability": 0.78125}, {"start": 1580.77, "end": 1581.05, "word": " said", "probability": 0.56591796875}, {"start": 1581.05, "end": 1581.79, "word": " a", "probability": 0.6142578125}, {"start": 1581.79, "end": 1582.29, "word": " citation", "probability": 0.802734375}], "temperature": 1.0}, {"id": 57, "seek": 161212, "start": 1584.4, "end": 1612.12, "text": " Catfish should be one of the top 2% in the weight. Assuming the weights of catfish are normally distributed, at what weight in bounds should the citation, the notation be established? So in this board, he asked about what's the value of x, for example.", "tokens": [9565, 11608, 820, 312, 472, 295, 264, 1192, 568, 4, 294, 264, 3364, 13, 6281, 24919, 264, 17443, 295, 3857, 11608, 366, 5646, 12631, 11, 412, 437, 3364, 294, 29905, 820, 264, 45590, 11, 264, 24657, 312, 7545, 30, 407, 294, 341, 3150, 11, 415, 2351, 466, 437, 311, 264, 2158, 295, 2031, 11, 337, 1365, 13], "avg_logprob": -0.24946119250922366, "compression_ratio": 1.4795321637426901, "no_speech_prob": 0.0, "words": [{"start": 1584.4, "end": 1585.16, "word": " Catfish", "probability": 0.55877685546875}, {"start": 1585.16, "end": 1585.92, "word": " should", "probability": 0.947265625}, {"start": 1585.92, "end": 1586.1, "word": " be", "probability": 0.9521484375}, {"start": 1586.1, "end": 1586.44, "word": " one", "probability": 0.90771484375}, {"start": 1586.44, "end": 1586.84, "word": " of", "probability": 0.96875}, {"start": 1586.84, "end": 1587.0, "word": " the", "probability": 0.92431640625}, {"start": 1587.0, "end": 1587.4, "word": " top", "probability": 0.94384765625}, {"start": 1587.4, "end": 1588.02, "word": " 2", "probability": 0.73046875}, {"start": 1588.02, "end": 1588.52, "word": "%", "probability": 0.93408203125}, {"start": 1588.52, "end": 1588.76, "word": " in", "probability": 0.91748046875}, {"start": 1588.76, "end": 1588.86, "word": " the", "probability": 0.77392578125}, {"start": 1588.86, "end": 1589.06, "word": " weight.", "probability": 0.9013671875}, {"start": 1591.64, "end": 1592.4, "word": " Assuming", "probability": 0.95849609375}, {"start": 1592.4, "end": 1592.62, "word": " the", "probability": 0.8984375}, {"start": 1592.62, "end": 1593.02, "word": " weights", "probability": 0.84423828125}, {"start": 1593.02, "end": 1593.42, "word": " of", "probability": 0.97021484375}, {"start": 1593.42, "end": 1593.86, "word": " catfish", "probability": 0.947265625}, {"start": 1593.86, "end": 1594.06, "word": " are", "probability": 0.93798828125}, {"start": 1594.06, "end": 1594.44, "word": " normally", "probability": 0.7978515625}, {"start": 1594.44, "end": 1595.02, "word": " distributed,", "probability": 0.8974609375}, {"start": 1596.54, "end": 1596.76, "word": " at", "probability": 0.92236328125}, {"start": 1596.76, "end": 1596.94, "word": " what", "probability": 0.9462890625}, {"start": 1596.94, "end": 1597.42, "word": " weight", "probability": 0.89013671875}, {"start": 1597.42, "end": 1598.24, "word": " in", "probability": 0.64501953125}, {"start": 1598.24, "end": 1598.66, "word": " bounds", "probability": 0.169189453125}, {"start": 1598.66, "end": 1599.56, "word": " should", "probability": 0.95068359375}, {"start": 1599.56, "end": 1599.74, "word": " the", "probability": 0.8671875}, {"start": 1599.74, "end": 1600.22, "word": " citation,", "probability": 0.84619140625}, {"start": 1601.58, "end": 1601.8, "word": " the", "probability": 0.81005859375}, {"start": 1601.8, "end": 1602.24, "word": " notation", "probability": 0.484619140625}, {"start": 1602.24, "end": 1602.62, "word": " be", "probability": 0.69677734375}, {"start": 1602.62, "end": 1603.68, "word": " established?", "probability": 0.91796875}, {"start": 1605.8, "end": 1606.56, "word": " So", "probability": 0.94970703125}, {"start": 1606.56, "end": 1606.9, "word": " in", "probability": 0.42724609375}, {"start": 1606.9, "end": 1607.24, "word": " this", "probability": 0.94384765625}, {"start": 1607.24, "end": 1608.9, "word": " board,", "probability": 0.75537109375}, {"start": 1609.04, "end": 1609.14, "word": " he", "probability": 0.892578125}, {"start": 1609.14, "end": 1609.3, "word": " asked", "probability": 0.42236328125}, {"start": 1609.3, "end": 1609.62, "word": " about", "probability": 0.8935546875}, {"start": 1609.62, "end": 1610.18, "word": " what's", "probability": 0.82373046875}, {"start": 1610.18, "end": 1610.32, "word": " the", "probability": 0.919921875}, {"start": 1610.32, "end": 1610.6, "word": " value", "probability": 0.97900390625}, {"start": 1610.6, "end": 1611.18, "word": " of", "probability": 0.363037109375}, {"start": 1611.18, "end": 1611.5, "word": " x,", "probability": 0.712890625}, {"start": 1611.62, "end": 1611.76, "word": " for", "probability": 0.9501953125}, {"start": 1611.76, "end": 1612.12, "word": " example.", "probability": 0.97216796875}], "temperature": 1.0}, {"id": 58, "seek": 164356, "start": 1617.16, "end": 1643.56, "text": " is greater than what value here. And this probability equals 2%. Because you said the citation catfish should be one of the top 2%. So the area in the right here, this area is 2%.", "tokens": [307, 5044, 813, 437, 2158, 510, 13, 400, 341, 8482, 6915, 568, 6856, 1436, 291, 848, 264, 45590, 3857, 11608, 820, 312, 472, 295, 264, 1192, 568, 6856, 407, 264, 1859, 294, 264, 558, 510, 11, 341, 1859, 307, 568, 6856], "avg_logprob": -0.23325892431395395, "compression_ratio": 1.3636363636363635, "no_speech_prob": 0.0, "words": [{"start": 1617.16, "end": 1617.36, "word": " is", "probability": 0.0919189453125}, {"start": 1617.36, "end": 1617.68, "word": " greater", "probability": 0.8818359375}, {"start": 1617.68, "end": 1618.18, "word": " than", "probability": 0.9423828125}, {"start": 1618.18, "end": 1620.26, "word": " what", "probability": 0.87109375}, {"start": 1620.26, "end": 1620.54, "word": " value", "probability": 0.95068359375}, {"start": 1620.54, "end": 1620.82, "word": " here.", "probability": 0.78759765625}, {"start": 1621.08, "end": 1621.48, "word": " And", "probability": 0.9404296875}, {"start": 1621.48, "end": 1621.68, "word": " this", "probability": 0.9375}, {"start": 1621.68, "end": 1622.12, "word": " probability", "probability": 0.90283203125}, {"start": 1622.12, "end": 1622.72, "word": " equals", "probability": 0.90087890625}, {"start": 1622.72, "end": 1624.76, "word": " 2%.", "probability": 0.809326171875}, {"start": 1624.76, "end": 1627.24, "word": " Because", "probability": 0.93603515625}, {"start": 1627.24, "end": 1627.58, "word": " you", "probability": 0.759765625}, {"start": 1627.58, "end": 1627.76, "word": " said", "probability": 0.7958984375}, {"start": 1627.76, "end": 1627.88, "word": " the", "probability": 0.56005859375}, {"start": 1627.88, "end": 1628.36, "word": " citation", "probability": 0.85986328125}, {"start": 1628.36, "end": 1629.08, "word": " catfish", "probability": 0.6478271484375}, {"start": 1629.08, "end": 1629.52, "word": " should", "probability": 0.951171875}, {"start": 1629.52, "end": 1629.84, "word": " be", "probability": 0.93017578125}, {"start": 1629.84, "end": 1630.28, "word": " one", "probability": 0.88037109375}, {"start": 1630.28, "end": 1630.54, "word": " of", "probability": 0.96630859375}, {"start": 1630.54, "end": 1630.7, "word": " the", "probability": 0.92333984375}, {"start": 1630.7, "end": 1630.96, "word": " top", "probability": 0.94287109375}, {"start": 1630.96, "end": 1631.7, "word": " 2%.", "probability": 0.95849609375}, {"start": 1631.7, "end": 1634.72, "word": " So", "probability": 0.951171875}, {"start": 1634.72, "end": 1634.94, "word": " the", "probability": 0.8583984375}, {"start": 1634.94, "end": 1635.42, "word": " area", "probability": 0.88671875}, {"start": 1635.42, "end": 1637.46, "word": " in", "probability": 0.287841796875}, {"start": 1637.46, "end": 1638.54, "word": " the", "probability": 0.828125}, {"start": 1638.54, "end": 1638.82, "word": " right", "probability": 0.923828125}, {"start": 1638.82, "end": 1639.2, "word": " here,", "probability": 0.8427734375}, {"start": 1640.48, "end": 1640.74, "word": " this", "probability": 0.9384765625}, {"start": 1640.74, "end": 1641.16, "word": " area", "probability": 0.89013671875}, {"start": 1641.16, "end": 1643.06, "word": " is", "probability": 0.82568359375}, {"start": 1643.06, "end": 1643.56, "word": " 2%.", "probability": 0.965087890625}], "temperature": 1.0}, {"id": 59, "seek": 165946, "start": 1646.0, "end": 1659.46, "text": " What's the value of x in this case? So here, the value of x greater than a equals 0.02, and we are looking for this value.", "tokens": [708, 311, 264, 2158, 295, 2031, 294, 341, 1389, 30, 407, 510, 11, 264, 2158, 295, 2031, 5044, 813, 257, 6915, 1958, 13, 12756, 11, 293, 321, 366, 1237, 337, 341, 2158, 13], "avg_logprob": -0.2828584681538975, "compression_ratio": 1.23, "no_speech_prob": 0.0, "words": [{"start": 1646.0, "end": 1646.6, "word": " What's", "probability": 0.6142578125}, {"start": 1646.6, "end": 1646.76, "word": " the", "probability": 0.92431640625}, {"start": 1646.76, "end": 1647.1, "word": " value", "probability": 0.97265625}, {"start": 1647.1, "end": 1648.34, "word": " of", "probability": 0.95751953125}, {"start": 1648.34, "end": 1648.78, "word": " x", "probability": 0.59619140625}, {"start": 1648.78, "end": 1649.3, "word": " in", "probability": 0.6943359375}, {"start": 1649.3, "end": 1649.52, "word": " this", "probability": 0.91748046875}, {"start": 1649.52, "end": 1649.9, "word": " case?", "probability": 0.8916015625}, {"start": 1651.98, "end": 1652.62, "word": " So", "probability": 0.81396484375}, {"start": 1652.62, "end": 1652.9, "word": " here,", "probability": 0.76806640625}, {"start": 1653.94, "end": 1654.08, "word": " the", "probability": 0.345703125}, {"start": 1654.08, "end": 1654.26, "word": " value", "probability": 0.1763916015625}, {"start": 1654.26, "end": 1654.52, "word": " of", "probability": 0.90869140625}, {"start": 1654.52, "end": 1654.72, "word": " x", "probability": 0.962890625}, {"start": 1654.72, "end": 1655.12, "word": " greater", "probability": 0.69384765625}, {"start": 1655.12, "end": 1655.44, "word": " than", "probability": 0.95703125}, {"start": 1655.44, "end": 1655.68, "word": " a", "probability": 0.58984375}, {"start": 1655.68, "end": 1656.16, "word": " equals", "probability": 0.81103515625}, {"start": 1656.16, "end": 1656.44, "word": " 0", "probability": 0.73681640625}, {"start": 1656.44, "end": 1657.14, "word": ".02,", "probability": 0.973388671875}, {"start": 1657.74, "end": 1658.14, "word": " and", "probability": 0.93212890625}, {"start": 1658.14, "end": 1658.28, "word": " we", "probability": 0.95263671875}, {"start": 1658.28, "end": 1658.42, "word": " are", "probability": 0.87451171875}, {"start": 1658.42, "end": 1658.64, "word": " looking", "probability": 0.912109375}, {"start": 1658.64, "end": 1658.92, "word": " for", "probability": 0.953125}, {"start": 1658.92, "end": 1659.14, "word": " this", "probability": 0.94677734375}, {"start": 1659.14, "end": 1659.46, "word": " value.", "probability": 0.9462890625}], "temperature": 1.0}, {"id": 60, "seek": 168909, "start": 1665.75, "end": 1689.09, "text": " gives the area to the left side. So this probability actually, the area to the right is 2%, so the area to the left is 98%. So this is the same as, as we know, the equal sign does not matter because we have continuous distribution.", "tokens": [2709, 264, 1859, 281, 264, 1411, 1252, 13, 407, 341, 8482, 767, 11, 264, 1859, 281, 264, 558, 307, 568, 8923, 370, 264, 1859, 281, 264, 1411, 307, 20860, 6856, 407, 341, 307, 264, 912, 382, 11, 382, 321, 458, 11, 264, 2681, 1465, 775, 406, 1871, 570, 321, 362, 10957, 7316, 13], "avg_logprob": -0.19285301201873356, "compression_ratio": 1.5466666666666666, "no_speech_prob": 0.0, "words": [{"start": 1665.75, "end": 1666.21, "word": " gives", "probability": 0.140380859375}, {"start": 1666.21, "end": 1666.37, "word": " the", "probability": 0.837890625}, {"start": 1666.37, "end": 1666.51, "word": " area", "probability": 0.8369140625}, {"start": 1666.51, "end": 1666.71, "word": " to", "probability": 0.94287109375}, {"start": 1666.71, "end": 1666.83, "word": " the", "probability": 0.9150390625}, {"start": 1666.83, "end": 1667.05, "word": " left", "probability": 0.93359375}, {"start": 1667.05, "end": 1667.37, "word": " side.", "probability": 0.76220703125}, {"start": 1668.49, "end": 1668.85, "word": " So", "probability": 0.85595703125}, {"start": 1668.85, "end": 1669.17, "word": " this", "probability": 0.71337890625}, {"start": 1669.17, "end": 1669.65, "word": " probability", "probability": 0.84130859375}, {"start": 1669.65, "end": 1670.39, "word": " actually,", "probability": 0.71533203125}, {"start": 1670.87, "end": 1672.59, "word": " the", "probability": 0.85888671875}, {"start": 1672.59, "end": 1672.79, "word": " area", "probability": 0.888671875}, {"start": 1672.79, "end": 1672.99, "word": " to", "probability": 0.95361328125}, {"start": 1672.99, "end": 1673.11, "word": " the", "probability": 0.9150390625}, {"start": 1673.11, "end": 1673.41, "word": " right", "probability": 0.90478515625}, {"start": 1673.41, "end": 1673.85, "word": " is", "probability": 0.92333984375}, {"start": 1673.85, "end": 1674.49, "word": " 2%,", "probability": 0.670654296875}, {"start": 1674.49, "end": 1674.97, "word": " so", "probability": 0.904296875}, {"start": 1674.97, "end": 1675.15, "word": " the", "probability": 0.9072265625}, {"start": 1675.15, "end": 1675.33, "word": " area", "probability": 0.884765625}, {"start": 1675.33, "end": 1675.53, "word": " to", "probability": 0.9609375}, {"start": 1675.53, "end": 1675.67, "word": " the", "probability": 0.91552734375}, {"start": 1675.67, "end": 1675.91, "word": " left", "probability": 0.94921875}, {"start": 1675.91, "end": 1677.17, "word": " is", "probability": 0.93505859375}, {"start": 1677.17, "end": 1678.01, "word": " 98%.", "probability": 0.924072265625}, {"start": 1678.01, "end": 1680.39, "word": " So", "probability": 0.90869140625}, {"start": 1680.39, "end": 1680.57, "word": " this", "probability": 0.8828125}, {"start": 1680.57, "end": 1680.67, "word": " is", "probability": 0.94677734375}, {"start": 1680.67, "end": 1680.81, "word": " the", "probability": 0.9111328125}, {"start": 1680.81, "end": 1681.03, "word": " same", "probability": 0.9189453125}, {"start": 1681.03, "end": 1681.41, "word": " as,", "probability": 0.96142578125}, {"start": 1684.61, "end": 1685.95, "word": " as", "probability": 0.92578125}, {"start": 1685.95, "end": 1686.09, "word": " we", "probability": 0.935546875}, {"start": 1686.09, "end": 1686.27, "word": " know,", "probability": 0.8857421875}, {"start": 1686.43, "end": 1686.53, "word": " the", "probability": 0.90869140625}, {"start": 1686.53, "end": 1686.75, "word": " equal", "probability": 0.8046875}, {"start": 1686.75, "end": 1687.03, "word": " sign", "probability": 0.8994140625}, {"start": 1687.03, "end": 1687.25, "word": " does", "probability": 0.90380859375}, {"start": 1687.25, "end": 1687.41, "word": " not", "probability": 0.95361328125}, {"start": 1687.41, "end": 1687.65, "word": " matter", "probability": 0.859375}, {"start": 1687.65, "end": 1687.93, "word": " because", "probability": 0.6884765625}, {"start": 1687.93, "end": 1688.09, "word": " we", "probability": 0.93017578125}, {"start": 1688.09, "end": 1688.19, "word": " have", "probability": 0.93994140625}, {"start": 1688.19, "end": 1688.67, "word": " continuous", "probability": 0.869140625}, {"start": 1688.67, "end": 1689.09, "word": " distribution.", "probability": 0.85693359375}], "temperature": 1.0}, {"id": 61, "seek": 171857, "start": 1692.05, "end": 1718.57, "text": " continuous distribution, so equal sign does not matter. So now, if you ask about P of X greater than a certain value equals a probability of, for example, 0.02, you have to find the probability to the left, which is 0.98, because our table gives the area to the left. 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So again, we have to look at the normal table, but backwards, because this value is given. 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So the answer might be your z-score could be 2.05 or 2.06. So again, in this case, the table does not give the exact. 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Or just take one of these. So either you can take 9798, which is closer to 98% than 9803, because it's three distant apart. So maybe we can take this value. Again, if you take the other one, you will be OK. 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So let's take the first value, for example. So my x equals mu, z is 2.05, times sigma, 0.8. 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So around 4.8. So your answer is 4.84. Now if you go back to the problem, and suppose you know the value of x.", "tokens": [708, 311, 428, 1867, 30, 805, 13, 17, 1804, 568, 935, 485, 407, 926, 1017, 13, 23, 13, 407, 428, 1867, 307, 1017, 13, 25494, 13, 823, 498, 291, 352, 646, 281, 264, 1154, 11, 293, 7297, 291, 458, 264, 2158, 295, 2031, 13], "avg_logprob": -0.20034721692403157, "compression_ratio": 1.2377049180327868, "no_speech_prob": 0.0, "words": [{"start": 1862.75, "end": 1863.11, "word": " What's", "probability": 0.732421875}, {"start": 1863.11, "end": 1863.25, "word": " your", "probability": 0.8955078125}, {"start": 1863.25, "end": 1863.61, "word": " answer?", "probability": 0.95263671875}, {"start": 1868.39, "end": 1869.11, "word": " 3", "probability": 0.79736328125}, {"start": 1869.11, "end": 1869.63, "word": ".2", "probability": 0.961669921875}, {"start": 1869.63, "end": 1870.23, "word": " plus", "probability": 0.73046875}, {"start": 1870.23, "end": 1870.57, "word": " 2", "probability": 0.81103515625}, {"start": 1870.57, "end": 1872.41, "word": " point...", "probability": 0.3919677734375}, {"start": 1872.41, "end": 1873.19, "word": " So", "probability": 0.428466796875}, {"start": 1873.19, "end": 1873.41, "word": " around", "probability": 0.568359375}, {"start": 1873.41, "end": 1873.71, "word": " 4", "probability": 0.9833984375}, {"start": 1873.71, "end": 1874.35, "word": ".8.", "probability": 0.98681640625}, {"start": 1876.13, "end": 1876.85, "word": " So", "probability": 0.9326171875}, {"start": 1876.85, "end": 1877.11, "word": " your", "probability": 0.83642578125}, {"start": 1877.11, "end": 1877.45, "word": " answer", "probability": 0.95703125}, {"start": 1877.45, "end": 1877.87, "word": " is", "probability": 0.9501953125}, {"start": 1877.87, "end": 1878.31, "word": " 4", "probability": 0.9755859375}, {"start": 1878.31, "end": 1878.89, "word": ".84.", "probability": 0.984375}, {"start": 1883.81, "end": 1884.53, "word": " Now", "probability": 0.94482421875}, {"start": 1884.53, "end": 1884.71, "word": " if", "probability": 0.6845703125}, {"start": 1884.71, "end": 1884.81, "word": " you", "probability": 0.95947265625}, {"start": 1884.81, "end": 1884.99, "word": " go", "probability": 0.9580078125}, {"start": 1884.99, "end": 1885.27, "word": " back", "probability": 0.8759765625}, {"start": 1885.27, "end": 1885.41, "word": " to", "probability": 0.94921875}, {"start": 1885.41, "end": 1885.49, "word": " the", "probability": 0.91552734375}, {"start": 1885.49, "end": 1885.83, "word": " problem,", "probability": 0.86865234375}, {"start": 1886.79, "end": 1887.19, "word": " and", "probability": 0.904296875}, {"start": 1887.19, "end": 1887.81, "word": " suppose", "probability": 0.91162109375}, {"start": 1887.81, "end": 1889.47, "word": " you", "probability": 0.8212890625}, {"start": 1889.47, "end": 1889.63, "word": " know", "probability": 0.89453125}, {"start": 1889.63, "end": 1889.79, "word": " the", "probability": 0.9189453125}, {"start": 1889.79, "end": 1890.15, "word": " value", "probability": 0.97314453125}, {"start": 1890.15, "end": 1890.49, "word": " of", "probability": 0.9677734375}, {"start": 1890.49, "end": 1890.83, "word": " x.", "probability": 0.4951171875}], "temperature": 1.0}, {"id": 68, "seek": 192363, "start": 1894.25, "end": 1923.63, "text": " So the probability of X. Double check to the answer. 4.84. Just check. V of X greater than this value should be", "tokens": [407, 264, 8482, 295, 1783, 13, 16633, 1520, 281, 264, 1867, 13, 1017, 13, 25494, 13, 1449, 1520, 13, 691, 295, 1783, 5044, 813, 341, 2158, 820, 312], "avg_logprob": -0.23356680417882986, "compression_ratio": 1.1313131313131313, "no_speech_prob": 0.0, "words": [{"start": 1894.25, "end": 1894.71, "word": " So", "probability": 0.759765625}, {"start": 1894.71, "end": 1894.95, "word": " the", "probability": 0.68408203125}, {"start": 1894.95, "end": 1895.33, "word": " probability", "probability": 0.951171875}, {"start": 1895.33, "end": 1895.61, "word": " of", "probability": 0.96142578125}, {"start": 1895.61, "end": 1896.01, "word": " X.", "probability": 0.77099609375}, {"start": 1902.99, "end": 1903.79, "word": " Double", "probability": 0.5556640625}, {"start": 1903.79, "end": 1904.17, "word": " check", "probability": 0.82861328125}, {"start": 1904.17, "end": 1904.59, "word": " to", "probability": 0.734375}, {"start": 1904.59, "end": 1904.71, "word": " the", "probability": 0.904296875}, {"start": 1904.71, "end": 1905.11, "word": " answer.", "probability": 0.958984375}, {"start": 1909.49, "end": 1910.29, "word": " 4", "probability": 0.6083984375}, {"start": 1910.29, "end": 1910.83, "word": ".84.", "probability": 0.923828125}, {"start": 1911.97, "end": 1912.43, "word": " Just", "probability": 0.78466796875}, {"start": 1912.43, "end": 1912.87, "word": " check.", "probability": 0.89208984375}, {"start": 1914.95, "end": 1915.17, "word": " V", "probability": 0.47314453125}, {"start": 1915.17, "end": 1915.33, "word": " of", "probability": 0.94091796875}, {"start": 1915.33, "end": 1915.71, "word": " X", "probability": 0.84228515625}, {"start": 1915.71, "end": 1917.03, "word": " greater", "probability": 0.70703125}, {"start": 1917.03, "end": 1917.35, "word": " than", "probability": 0.9482421875}, {"start": 1917.35, "end": 1917.61, "word": " this", "probability": 0.9521484375}, {"start": 1917.61, "end": 1918.01, "word": " value", "probability": 0.97705078125}, {"start": 1918.01, "end": 1923.29, "word": " should", "probability": 0.78759765625}, {"start": 1923.29, "end": 1923.63, "word": " be", "probability": 0.95458984375}], "temperature": 1.0}, {"id": 69, "seek": 195058, "start": 1924.24, "end": 1950.58, "text": " Two percent. Two percent. So the probability of X greater than this value should be equal to one zero. So this problem is called backward normal calculation because here first step we find the value of this score corresponding to this probability. 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So my value here should be to the right. Because it says greater than A is just 2%. 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Above what weight? Do 89.8% of the weights care? 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X minus. And 98, 89, sorry, 89. So this is a percentage he's looking for. 89.8%. Now what's the value of A? Or above what weight? 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Now, A should be to the right or to the left side? Imagine A in the right side here. Do you think the area above A is 89%? 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So this is your 8. Now, this area makes sense that it's above 0.5. It's 0.8980. Now, B of X greater than equals this value. And again, the table gives the area to the left. So this is actually", "tokens": [407, 309, 820, 312, 264, 661, 1252, 13, 407, 341, 307, 428, 1649, 13, 823, 11, 341, 1859, 1669, 2020, 300, 309, 311, 3673, 1958, 13, 20, 13, 467, 311, 1958, 13, 21115, 4702, 13, 823, 11, 363, 295, 1783, 5044, 813, 6915, 341, 2158, 13, 400, 797, 11, 264, 3199, 2709, 264, 1859, 281, 264, 1411, 13, 407, 341, 307, 767], "avg_logprob": -0.2021329417115166, "compression_ratio": 1.4240506329113924, "no_speech_prob": 0.0, "words": [{"start": 2121.05, "end": 2121.35, "word": " So", "probability": 0.6787109375}, {"start": 2121.35, "end": 2121.49, "word": " it", "probability": 0.78515625}, {"start": 2121.49, "end": 2121.67, "word": " should", "probability": 0.96044921875}, {"start": 2121.67, "end": 2121.99, "word": " be", "probability": 0.953125}, {"start": 2121.99, "end": 2123.59, "word": " the", "probability": 0.82275390625}, {"start": 2123.59, "end": 2123.81, "word": " other", "probability": 0.89208984375}, {"start": 2123.81, "end": 2124.25, "word": " side.", "probability": 0.8662109375}, {"start": 2126.77, "end": 2127.27, "word": " So", "probability": 0.83447265625}, {"start": 2127.27, "end": 2127.49, "word": " this", "probability": 0.8974609375}, {"start": 2127.49, "end": 2127.59, "word": " is", "probability": 0.93896484375}, {"start": 2127.59, "end": 2127.75, "word": " your", "probability": 0.896484375}, {"start": 2127.75, "end": 2128.03, "word": " 8.", "probability": 0.5107421875}, {"start": 2129.43, "end": 2129.69, "word": " Now,", "probability": 0.94287109375}, {"start": 2129.77, "end": 2129.95, "word": " this", "probability": 0.9267578125}, {"start": 2129.95, "end": 2130.31, "word": " area", "probability": 0.91259765625}, {"start": 2130.31, "end": 2131.15, "word": " makes", "probability": 0.483154296875}, {"start": 2131.15, "end": 2131.83, "word": " sense", "probability": 0.77978515625}, {"start": 2131.83, "end": 2132.17, "word": " that", "probability": 0.90966796875}, {"start": 2132.17, "end": 2133.29, "word": " it's", "probability": 0.953369140625}, {"start": 2133.29, "end": 2133.53, "word": " above", "probability": 0.9306640625}, {"start": 2133.53, "end": 2133.85, "word": " 0", "probability": 0.63671875}, {"start": 2133.85, "end": 2134.17, "word": ".5.", "probability": 0.99169921875}, {"start": 2134.43, "end": 2134.75, "word": " It's", "probability": 0.935302734375}, {"start": 2134.75, "end": 2135.03, "word": " 0", "probability": 0.96923828125}, {"start": 2135.03, "end": 2136.69, "word": ".8980.", "probability": 0.91845703125}, {"start": 2138.75, "end": 2139.31, "word": " Now,", "probability": 0.955078125}, {"start": 2139.65, "end": 2139.81, "word": " B", "probability": 0.27392578125}, {"start": 2139.81, "end": 2139.93, "word": " of", "probability": 0.69287109375}, {"start": 2139.93, "end": 2140.19, "word": " X", "probability": 0.6103515625}, {"start": 2140.19, "end": 2140.55, "word": " greater", "probability": 0.87109375}, {"start": 2140.55, "end": 2140.79, "word": " than", "probability": 0.9501953125}, {"start": 2140.79, "end": 2141.05, "word": " equals", "probability": 0.501953125}, {"start": 2141.05, "end": 2141.49, "word": " this", "probability": 0.94189453125}, {"start": 2141.49, "end": 2141.83, "word": " value.", "probability": 0.97265625}, {"start": 2142.59, "end": 2142.85, "word": " And", "probability": 0.9384765625}, {"start": 2142.85, "end": 2143.09, "word": " again,", "probability": 0.87451171875}, {"start": 2143.15, "end": 2143.27, "word": " the", "probability": 0.9208984375}, {"start": 2143.27, "end": 2143.65, "word": " table", "probability": 0.8828125}, {"start": 2143.65, "end": 2144.93, "word": " gives", "probability": 0.90478515625}, {"start": 2144.93, "end": 2145.13, "word": " the", "probability": 0.9130859375}, {"start": 2145.13, "end": 2145.35, "word": " area", "probability": 0.884765625}, {"start": 2145.35, "end": 2145.55, "word": " to", "probability": 0.962890625}, {"start": 2145.55, "end": 2145.69, "word": " the", "probability": 0.91455078125}, {"start": 2145.69, "end": 2145.97, "word": " left.", "probability": 0.951171875}, {"start": 2146.71, "end": 2146.99, "word": " So", "probability": 0.89697265625}, {"start": 2146.99, "end": 2147.31, "word": " this", "probability": 0.92724609375}, {"start": 2147.31, "end": 2147.47, "word": " is", "probability": 0.94140625}, {"start": 2147.47, "end": 2148.07, "word": " actually", "probability": 0.884765625}], "temperature": 1.0}, {"id": 77, "seek": 217780, "start": 2148.96, "end": 2177.8, "text": " X less than A, 1 minus this value, equals 0.1020. Now go back to the normal table. Here it's 0.1020. So it should be negative. I mean, your z-scope should be negative.", "tokens": [1783, 1570, 813, 316, 11, 502, 3175, 341, 2158, 11, 6915, 1958, 13, 3279, 2009, 13, 823, 352, 646, 281, 264, 2710, 3199, 13, 1692, 309, 311, 1958, 13, 3279, 2009, 13, 407, 309, 820, 312, 3671, 13, 286, 914, 11, 428, 710, 12, 4417, 1114, 820, 312, 3671, 13], "avg_logprob": -0.22380514179959016, "compression_ratio": 1.3228346456692914, "no_speech_prob": 0.0, "words": [{"start": 2148.96, "end": 2149.4, "word": " X", "probability": 0.381103515625}, {"start": 2149.4, "end": 2150.88, "word": " less", "probability": 0.80078125}, {"start": 2150.88, "end": 2151.18, "word": " than", "probability": 0.94873046875}, {"start": 2151.18, "end": 2151.5, "word": " A,", "probability": 0.381591796875}, {"start": 2152.0, "end": 2153.08, "word": " 1", "probability": 0.666015625}, {"start": 2153.08, "end": 2153.46, "word": " minus", "probability": 0.98388671875}, {"start": 2153.46, "end": 2153.74, "word": " this", "probability": 0.92431640625}, {"start": 2153.74, "end": 2154.16, "word": " value,", "probability": 0.97265625}, {"start": 2155.18, "end": 2155.54, "word": " equals", "probability": 0.482177734375}, {"start": 2155.54, "end": 2155.82, "word": " 0", "probability": 0.8251953125}, {"start": 2155.82, "end": 2156.6, "word": ".1020.", "probability": 0.9065755208333334}, {"start": 2161.48, "end": 2162.2, "word": " Now", "probability": 0.85107421875}, {"start": 2162.2, "end": 2164.26, "word": " go", "probability": 0.42724609375}, {"start": 2164.26, "end": 2164.66, "word": " back", "probability": 0.87646484375}, {"start": 2164.66, "end": 2168.76, "word": " to", "probability": 0.900390625}, {"start": 2168.76, "end": 2168.9, "word": " the", "probability": 0.91650390625}, {"start": 2168.9, "end": 2169.2, "word": " normal", "probability": 0.90234375}, {"start": 2169.2, "end": 2169.58, "word": " table.", "probability": 0.88134765625}, {"start": 2170.52, "end": 2170.8, "word": " Here", "probability": 0.85595703125}, {"start": 2170.8, "end": 2171.06, "word": " it's", "probability": 0.844482421875}, {"start": 2171.06, "end": 2171.28, "word": " 0", "probability": 0.98046875}, {"start": 2171.28, "end": 2172.12, "word": ".1020.", "probability": 0.8680013020833334}, {"start": 2173.42, "end": 2174.04, "word": " So", "probability": 0.96337890625}, {"start": 2174.04, "end": 2174.2, "word": " it", "probability": 0.428466796875}, {"start": 2174.2, "end": 2174.4, "word": " should", "probability": 0.966796875}, {"start": 2174.4, "end": 2174.56, "word": " be", "probability": 0.9521484375}, {"start": 2174.56, "end": 2174.88, "word": " negative.", "probability": 0.94384765625}, {"start": 2175.72, "end": 2175.84, "word": " I", "probability": 0.84716796875}, {"start": 2175.84, "end": 2175.96, "word": " mean,", "probability": 0.9658203125}, {"start": 2175.98, "end": 2176.16, "word": " your", "probability": 0.7587890625}, {"start": 2176.16, "end": 2176.32, "word": " z", "probability": 0.51904296875}, {"start": 2176.32, "end": 2176.74, "word": "-scope", "probability": 0.85498046875}, {"start": 2176.74, "end": 2177.32, "word": " should", "probability": 0.96875}, {"start": 2177.32, "end": 2177.5, "word": " be", "probability": 0.951171875}, {"start": 2177.5, "end": 2177.8, "word": " negative.", "probability": 0.94482421875}], "temperature": 1.0}, {"id": 78, "seek": 221123, "start": 2182.0, "end": 2211.24, "text": " Now look at 0.102. Exactly this value. 0.102 is minus 1.2 up to 7. So minus 1.27. Minus 1.2.", "tokens": [823, 574, 412, 1958, 13, 3279, 17, 13, 7587, 341, 2158, 13, 1958, 13, 3279, 17, 307, 3175, 502, 13, 17, 493, 281, 1614, 13, 407, 3175, 502, 13, 10076, 13, 2829, 301, 502, 13, 17, 13], "avg_logprob": -0.16930509162576576, "compression_ratio": 1.1625, "no_speech_prob": 0.0, "words": [{"start": 2182.0, "end": 2182.24, "word": " Now", "probability": 0.86669921875}, {"start": 2182.24, "end": 2182.46, "word": " look", "probability": 0.77490234375}, {"start": 2182.46, "end": 2182.64, "word": " at", "probability": 0.9677734375}, {"start": 2182.64, "end": 2182.92, "word": " 0", "probability": 0.408203125}, {"start": 2182.92, "end": 2185.64, "word": ".102.", "probability": 0.9462890625}, {"start": 2188.56, "end": 2189.6, "word": " Exactly", "probability": 0.69580078125}, {"start": 2189.6, "end": 2189.94, "word": " this", "probability": 0.93017578125}, {"start": 2189.94, "end": 2190.22, "word": " value.", "probability": 0.96044921875}, {"start": 2191.54, "end": 2192.26, "word": " 0", "probability": 0.947265625}, {"start": 2192.26, "end": 2192.94, "word": ".102", "probability": 0.97802734375}, {"start": 2192.94, "end": 2193.14, "word": " is", "probability": 0.44677734375}, {"start": 2193.14, "end": 2193.84, "word": " minus", "probability": 0.94091796875}, {"start": 2193.84, "end": 2194.9, "word": " 1", "probability": 0.94580078125}, {"start": 2194.9, "end": 2195.6, "word": ".2", "probability": 0.97900390625}, {"start": 2195.6, "end": 2196.58, "word": " up", "probability": 0.88916015625}, {"start": 2196.58, "end": 2196.8, "word": " to", "probability": 0.970703125}, {"start": 2196.8, "end": 2197.1, "word": " 7.", "probability": 0.65283203125}, {"start": 2198.18, "end": 2198.68, "word": " So", "probability": 0.55810546875}, {"start": 2198.68, "end": 2198.96, "word": " minus", "probability": 0.84228515625}, {"start": 2198.96, "end": 2199.28, "word": " 1", "probability": 0.99267578125}, {"start": 2199.28, "end": 2199.9, "word": ".27.", "probability": 0.98486328125}, {"start": 2209.12, "end": 2210.16, "word": " Minus", "probability": 0.970458984375}, {"start": 2210.16, "end": 2210.68, "word": " 1", "probability": 0.99462890625}, {"start": 2210.68, "end": 2211.24, "word": ".2.", "probability": 0.986328125}], "temperature": 1.0}, {"id": 79, "seek": 223996, "start": 2212.4, "end": 2239.96, "text": " All the way up to 7 is 0.102. So the corresponding z-score is minus 1.17. Now x again equals mu plus z sigma. Mu is 3.2 plus z is negative 1.17 times sigma.", "tokens": [1057, 264, 636, 493, 281, 1614, 307, 1958, 13, 3279, 17, 13, 407, 264, 11760, 710, 12, 4417, 418, 307, 3175, 502, 13, 7773, 13, 823, 2031, 797, 6915, 2992, 1804, 710, 12771, 13, 15601, 307, 805, 13, 17, 1804, 710, 307, 3671, 502, 13, 7773, 1413, 12771, 13], "avg_logprob": -0.19484375566244125, "compression_ratio": 1.256, "no_speech_prob": 0.0, "words": [{"start": 2212.4, "end": 2212.7, "word": " All", "probability": 0.60888671875}, {"start": 2212.7, "end": 2212.86, "word": " the", "probability": 0.916015625}, {"start": 2212.86, "end": 2212.98, "word": " way", "probability": 0.9560546875}, {"start": 2212.98, "end": 2213.14, "word": " up", "probability": 0.9423828125}, {"start": 2213.14, "end": 2213.28, "word": " to", "probability": 0.95751953125}, {"start": 2213.28, "end": 2213.56, "word": " 7", "probability": 0.499755859375}, {"start": 2213.56, "end": 2214.26, "word": " is", "probability": 0.67431640625}, {"start": 2214.26, "end": 2214.46, "word": " 0", "probability": 0.7177734375}, {"start": 2214.46, "end": 2215.2, "word": ".102.", "probability": 0.9578450520833334}, {"start": 2216.28, "end": 2216.98, "word": " So", "probability": 0.95361328125}, {"start": 2216.98, "end": 2217.9, "word": " the", "probability": 0.68505859375}, {"start": 2217.9, "end": 2218.44, "word": " corresponding", "probability": 0.8408203125}, {"start": 2218.44, "end": 2218.72, "word": " z", "probability": 0.6474609375}, {"start": 2218.72, "end": 2219.08, "word": "-score", "probability": 0.7659505208333334}, {"start": 2219.08, "end": 2219.52, "word": " is", "probability": 0.94189453125}, {"start": 2219.52, "end": 2219.84, "word": " minus", "probability": 0.56201171875}, {"start": 2219.84, "end": 2220.08, "word": " 1", "probability": 0.9716796875}, {"start": 2220.08, "end": 2220.7, "word": ".17.", "probability": 0.900146484375}, {"start": 2222.52, "end": 2223.3, "word": " Now", "probability": 0.9619140625}, {"start": 2223.3, "end": 2223.68, "word": " x", "probability": 0.53271484375}, {"start": 2223.68, "end": 2224.16, "word": " again", "probability": 0.76025390625}, {"start": 2224.16, "end": 2224.96, "word": " equals", "probability": 0.89453125}, {"start": 2224.96, "end": 2225.16, "word": " mu", "probability": 0.71044921875}, {"start": 2225.16, "end": 2225.42, "word": " plus", "probability": 0.95849609375}, {"start": 2225.42, "end": 2225.64, "word": " z", "probability": 0.59814453125}, {"start": 2225.64, "end": 2225.98, "word": " sigma.", "probability": 0.80615234375}, {"start": 2230.28, "end": 2231.06, "word": " Mu", "probability": 0.736328125}, {"start": 2231.06, "end": 2231.58, "word": " is", "probability": 0.91162109375}, {"start": 2231.58, "end": 2231.82, "word": " 3", "probability": 0.94287109375}, {"start": 2231.82, "end": 2232.34, "word": ".2", "probability": 0.959228515625}, {"start": 2232.34, "end": 2233.14, "word": " plus", "probability": 0.7509765625}, {"start": 2233.14, "end": 2235.62, "word": " z", "probability": 0.61962890625}, {"start": 2235.62, "end": 2235.76, "word": " is", "probability": 0.89892578125}, {"start": 2235.76, "end": 2236.06, "word": " negative", "probability": 0.92919921875}, {"start": 2236.06, "end": 2236.36, "word": " 1", "probability": 0.98193359375}, {"start": 2236.36, "end": 2236.98, "word": ".17", "probability": 0.994873046875}, {"start": 2236.98, "end": 2239.56, "word": " times", "probability": 0.654296875}, {"start": 2239.56, "end": 2239.96, "word": " sigma.", "probability": 0.92431640625}], "temperature": 1.0}, {"id": 80, "seek": 225651, "start": 2244.25, "end": 2256.51, "text": " So it's equal to 3.2 minus 127 times 0.3. By calculator, you'll get the final result.", "tokens": [407, 309, 311, 2681, 281, 805, 13, 17, 3175, 47561, 1413, 1958, 13, 18, 13, 3146, 24993, 11, 291, 603, 483, 264, 2572, 1874, 13], "avg_logprob": -0.26998196083765763, "compression_ratio": 1.0, "no_speech_prob": 0.0, "words": [{"start": 2244.25, "end": 2245.21, "word": " So", "probability": 0.1641845703125}, {"start": 2245.21, "end": 2246.17, "word": " it's", "probability": 0.826904296875}, {"start": 2246.17, "end": 2246.55, "word": " equal", "probability": 0.8974609375}, {"start": 2246.55, "end": 2247.05, "word": " to", "probability": 0.97021484375}, {"start": 2247.05, "end": 2247.41, "word": " 3", "probability": 0.84130859375}, {"start": 2247.41, "end": 2247.91, "word": ".2", "probability": 0.985107421875}, {"start": 2247.91, "end": 2249.29, "word": " minus", "probability": 0.96435546875}, {"start": 2249.29, "end": 2251.05, "word": " 127", "probability": 0.72265625}, {"start": 2251.05, "end": 2251.69, "word": " times", "probability": 0.89697265625}, {"start": 2251.69, "end": 2251.95, "word": " 0", "probability": 0.50830078125}, {"start": 2251.95, "end": 2252.25, "word": ".3.", "probability": 0.853515625}, {"start": 2253.91, "end": 2254.39, "word": " By", "probability": 0.94775390625}, {"start": 2254.39, "end": 2254.89, "word": " calculator,", "probability": 0.65234375}, {"start": 2255.01, "end": 2255.15, "word": " you'll", "probability": 0.6572265625}, {"start": 2255.15, "end": 2255.41, "word": " get", "probability": 0.94189453125}, {"start": 2255.41, "end": 2255.69, "word": " the", "probability": 0.90771484375}, {"start": 2255.69, "end": 2256.09, "word": " final", "probability": 0.9501953125}, {"start": 2256.09, "end": 2256.51, "word": " result.", "probability": 0.9248046875}], "temperature": 1.0}, {"id": 81, "seek": 229602, "start": 2271.12, "end": 2296.02, "text": " If the probability is smaller than 0.5, then this one is negative. Go to the other one. If the probability is above 0.5, then use the positive z-score. So what's the answer? 2.18. Be careful. 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In this case, the value of A, for example, is located in the upper tail here. 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So you have U of X greater than equal this value, which is the score in this case minus 1.17. So the corresponding guess score actually is 2.24. So this is the weight that 89.8% of the weights are above it.", "tokens": [467, 311, 510, 544, 813, 1958, 13, 20, 820, 312, 322, 264, 661, 1252, 13, 407, 291, 362, 624, 295, 1783, 5044, 813, 2681, 341, 2158, 11, 597, 307, 264, 6175, 294, 341, 1389, 3175, 502, 13, 7773, 13, 407, 264, 11760, 2041, 6175, 767, 307, 568, 13, 7911, 13, 407, 341, 307, 264, 3364, 300, 31877, 13, 23, 4, 295, 264, 17443, 366, 3673, 309, 13], "avg_logprob": -0.21817554467741182, "compression_ratio": 1.452513966480447, "no_speech_prob": 0.0, "words": [{"start": 2328.61, "end": 2329.21, "word": " It's", "probability": 0.50543212890625}, {"start": 2329.21, "end": 2329.37, "word": " here", "probability": 0.646484375}, {"start": 2329.37, "end": 2329.63, "word": " more", "probability": 0.6943359375}, {"start": 2329.63, "end": 2329.81, "word": " than", "probability": 0.9365234375}, {"start": 2329.81, "end": 2330.07, "word": " 0", "probability": 0.40576171875}, {"start": 2330.07, "end": 2330.35, "word": ".5", 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2352.61, "end": 2353.47, "word": " are", "probability": 0.93359375}, {"start": 2353.47, "end": 2353.79, "word": " above", "probability": 0.947265625}, {"start": 2353.79, "end": 2354.11, "word": " it.", "probability": 0.947265625}], "temperature": 1.0}, {"id": 84, "seek": 238169, "start": 2355.77, "end": 2381.69, "text": " So around 90% of the catch fish have weights above this value. So around 2 pounds. So around 90% of the weights are above 2.18 pounds. Maybe this is one of the most important questions in this chapter. Any question?", "tokens": [407, 926, 4289, 4, 295, 264, 3745, 3506, 362, 17443, 3673, 341, 2158, 13, 407, 926, 568, 8319, 13, 407, 926, 4289, 4, 295, 264, 17443, 366, 3673, 568, 13, 6494, 8319, 13, 2704, 341, 307, 472, 295, 264, 881, 1021, 1651, 294, 341, 7187, 13, 2639, 1168, 30], "avg_logprob": -0.13859375327825546, "compression_ratio": 1.5766423357664234, "no_speech_prob": 0.0, "words": [{"start": 2355.77, "end": 2356.07, "word": " So", "probability": 0.86669921875}, {"start": 2356.07, "end": 2356.41, "word": " around", "probability": 0.80224609375}, {"start": 2356.41, "end": 2356.91, "word": " 90", "probability": 0.97021484375}, {"start": 2356.91, "end": 2357.23, "word": "%", "probability": 0.83251953125}, {"start": 2357.23, "end": 2357.81, "word": " of", "probability": 0.9677734375}, {"start": 2357.81, "end": 2357.97, "word": " the", "probability": 0.916015625}, {"start": 2357.97, "end": 2358.29, "word": " catch", "probability": 0.54150390625}, {"start": 2358.29, "end": 2358.77, "word": " fish", "probability": 0.9072265625}, {"start": 2358.77, "end": 2360.63, "word": " have", "probability": 0.76611328125}, {"start": 2360.63, "end": 2361.05, "word": " weights", "probability": 0.8994140625}, {"start": 2361.05, "end": 2361.43, "word": " above", "probability": 0.95556640625}, {"start": 2361.43, "end": 2362.63, "word": " this", "probability": 0.9267578125}, {"start": 2362.63, "end": 2363.03, "word": " value.", "probability": 0.97412109375}, {"start": 2364.53, "end": 2365.35, "word": " So", "probability": 0.6123046875}, {"start": 2365.35, "end": 2365.59, "word": " around", "probability": 0.79833984375}, {"start": 2365.59, "end": 2365.75, "word": " 2", "probability": 0.64453125}, {"start": 2365.75, "end": 2366.37, "word": " pounds.", "probability": 0.7373046875}, {"start": 2367.05, "end": 2367.45, "word": " So", "probability": 0.90869140625}, {"start": 2367.45, "end": 2367.69, "word": " around", "probability": 0.87158203125}, {"start": 2367.69, "end": 2367.97, "word": " 90", "probability": 0.98388671875}, {"start": 2367.97, "end": 2368.29, "word": "%", "probability": 0.98974609375}, {"start": 2368.29, "end": 2368.59, "word": " of", "probability": 0.9697265625}, {"start": 2368.59, "end": 2368.73, "word": " the", "probability": 0.89111328125}, {"start": 2368.73, "end": 2369.05, "word": " weights", "probability": 0.85302734375}, {"start": 2369.05, "end": 2371.17, "word": " are", "probability": 0.931640625}, {"start": 2371.17, "end": 2371.75, "word": " above", "probability": 0.9609375}, {"start": 2371.75, "end": 2372.19, "word": " 2", "probability": 0.94775390625}, {"start": 2372.19, "end": 2372.71, "word": ".18", "probability": 0.9091796875}, {"start": 2372.71, "end": 2373.13, "word": " pounds.", "probability": 0.79736328125}, {"start": 2375.45, "end": 2375.93, "word": " Maybe", "probability": 0.85546875}, {"start": 2375.93, "end": 2376.19, "word": " this", "probability": 0.7666015625}, {"start": 2376.19, "end": 2376.27, "word": " is", "probability": 0.861328125}, {"start": 2376.27, "end": 2376.43, "word": " one", "probability": 0.92578125}, {"start": 2376.43, "end": 2376.55, "word": " of", "probability": 0.96484375}, {"start": 2376.55, "end": 2376.69, "word": " the", "probability": 0.91455078125}, {"start": 2376.69, "end": 2377.07, "word": " most", "probability": 0.90380859375}, {"start": 2377.07, "end": 2377.69, "word": " important", "probability": 0.86279296875}, {"start": 2377.69, "end": 2378.35, "word": " questions", "probability": 0.94091796875}, {"start": 2378.35, "end": 2378.59, "word": " in", "probability": 0.93115234375}, {"start": 2378.59, "end": 2379.01, "word": " this", "probability": 0.943359375}, {"start": 2379.01, "end": 2379.45, "word": " chapter.", "probability": 0.85302734375}, {"start": 2380.47, "end": 2381.27, "word": " Any", "probability": 0.90625}, {"start": 2381.27, "end": 2381.69, "word": " question?", "probability": 0.7109375}], "temperature": 1.0}, {"id": 85, "seek": 241267, "start": 2391.66, "end": 2412.68, "text": " The last part here, for the same problem he asked about, what's the probability that a randomly selected fish will weigh less than 2.2 pounds? I think straightforward. We did similar in part A.", "tokens": [440, 1036, 644, 510, 11, 337, 264, 912, 1154, 415, 2351, 466, 11, 437, 311, 264, 8482, 300, 257, 16979, 8209, 3506, 486, 13843, 1570, 813, 568, 13, 17, 8319, 30, 286, 519, 15325, 13, 492, 630, 2531, 294, 644, 316, 13], "avg_logprob": -0.21584302949350934, "compression_ratio": 1.3197278911564625, "no_speech_prob": 0.0, "words": [{"start": 2391.6600000000003, "end": 2392.34, "word": " The", "probability": 0.7353515625}, {"start": 2392.34, "end": 2392.68, "word": " last", "probability": 0.873046875}, {"start": 2392.68, "end": 2393.12, "word": " part", "probability": 0.8984375}, {"start": 2393.12, "end": 2393.46, "word": " here,", "probability": 0.82763671875}, {"start": 2393.74, "end": 2393.88, "word": " for", "probability": 0.9208984375}, {"start": 2393.88, "end": 2394.04, "word": " the", "probability": 0.921875}, {"start": 2394.04, "end": 2394.38, "word": " same", "probability": 0.884765625}, {"start": 2394.38, "end": 2396.96, "word": " problem", "probability": 0.76318359375}, {"start": 2396.96, "end": 2397.3, "word": " he", "probability": 0.32958984375}, {"start": 2397.3, "end": 2397.58, "word": " asked", "probability": 0.5390625}, {"start": 2397.58, "end": 2398.08, "word": " about,", "probability": 0.91357421875}, {"start": 2399.84, "end": 2400.14, "word": " what's", "probability": 0.838623046875}, {"start": 2400.14, "end": 2400.24, "word": " the", "probability": 0.92431640625}, {"start": 2400.24, "end": 2400.68, "word": " probability", "probability": 0.96044921875}, {"start": 2400.68, "end": 2401.12, "word": " that", "probability": 0.9365234375}, {"start": 2401.12, "end": 2401.4, "word": " a", "probability": 0.96533203125}, {"start": 2401.4, "end": 2401.7, "word": " randomly", "probability": 0.78125}, {"start": 2401.7, "end": 2402.32, "word": " selected", "probability": 0.85498046875}, {"start": 2402.32, "end": 2402.7, "word": " fish", "probability": 0.75537109375}, {"start": 2402.7, "end": 2402.92, "word": " will", "probability": 0.88525390625}, {"start": 2402.92, "end": 2403.18, "word": " weigh", "probability": 0.87548828125}, {"start": 2403.18, "end": 2403.7, "word": " less", "probability": 0.9462890625}, {"start": 2403.7, "end": 2403.92, "word": " than", "probability": 0.947265625}, {"start": 2403.92, "end": 2404.14, "word": " 2", "probability": 0.9560546875}, {"start": 2404.14, "end": 2404.5, "word": ".2", "probability": 0.986572265625}, {"start": 2404.5, "end": 2404.84, "word": " pounds?", "probability": 0.890625}, {"start": 2405.66, "end": 2406.12, "word": " I", "probability": 0.96337890625}, {"start": 2406.12, "end": 2406.28, "word": " think", "probability": 0.916015625}, {"start": 2406.28, "end": 2406.86, "word": " straightforward.", "probability": 0.65185546875}, {"start": 2408.98, "end": 2409.38, "word": " We", "probability": 0.484130859375}, {"start": 2409.38, "end": 2409.68, "word": " did", "probability": 0.94873046875}, {"start": 2409.68, "end": 2410.56, "word": " similar", "probability": 0.81982421875}, {"start": 2410.56, "end": 2412.2, "word": " in", "probability": 0.8916015625}, {"start": 2412.2, "end": 2412.44, "word": " part", "probability": 0.71337890625}, {"start": 2412.44, "end": 2412.68, "word": " A.", "probability": 0.74072265625}], "temperature": 1.0}, {"id": 86, "seek": 244342, "start": 2414.9, "end": 2443.42, "text": " So B of X less than 0.2. So he's looking for this probability, which is straightforward one. This score, 3.2 minus, I'm sorry, it's 2.2 minus minus. It's 2.2 minus 3.2 divided by sigma.", "tokens": [407, 363, 295, 1783, 1570, 813, 1958, 13, 17, 13, 407, 415, 311, 1237, 337, 341, 8482, 11, 597, 307, 15325, 472, 13, 639, 6175, 11, 805, 13, 17, 3175, 11, 286, 478, 2597, 11, 309, 311, 568, 13, 17, 3175, 3175, 13, 467, 311, 568, 13, 17, 3175, 805, 13, 17, 6666, 538, 12771, 13], "avg_logprob": -0.22793311194369667, "compression_ratio": 1.3285714285714285, "no_speech_prob": 0.0, "words": [{"start": 2414.9, "end": 2415.24, "word": " So", "probability": 0.75830078125}, {"start": 2415.24, "end": 2415.4, "word": " B", "probability": 0.335693359375}, {"start": 2415.4, "end": 2415.56, "word": " of", "probability": 0.54931640625}, {"start": 2415.56, "end": 2415.88, "word": " X", "probability": 0.81005859375}, {"start": 2415.88, "end": 2417.7, "word": " less", "probability": 0.62548828125}, {"start": 2417.7, "end": 2417.94, "word": " than", "probability": 0.9296875}, {"start": 2417.94, "end": 2418.22, "word": " 0", "probability": 0.5166015625}, 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It's negative 1.15 minus 1.2 up to 5. So 0.1056. So the answer is around 10% of the catfish", "tokens": [823, 11, 294, 341, 1389, 11, 321, 362, 281, 764, 264, 3671, 710, 13, 467, 311, 3671, 502, 13, 5211, 3175, 502, 13, 17, 493, 281, 1025, 13, 407, 1958, 13, 3279, 18317, 13, 407, 264, 1867, 307, 926, 1266, 4, 295, 264, 3857, 11608], "avg_logprob": -0.17663043154322583, "compression_ratio": 1.1932773109243697, "no_speech_prob": 0.0, "words": [{"start": 2474.87, "end": 2475.17, "word": " Now,", "probability": 0.8818359375}, {"start": 2475.27, "end": 2475.35, "word": " in", "probability": 0.94091796875}, {"start": 2475.35, "end": 2475.51, "word": " this", "probability": 0.9482421875}, {"start": 2475.51, "end": 2475.69, "word": " case,", "probability": 0.91796875}, {"start": 2475.79, "end": 2475.83, "word": " we", "probability": 0.89111328125}, {"start": 2475.83, "end": 2475.97, "word": " have", "probability": 0.92333984375}, {"start": 2475.97, "end": 2476.11, "word": " to", "probability": 0.96142578125}, {"start": 2476.11, "end": 2476.39, "word": " use", "probability": 0.859375}, {"start": 2476.39, "end": 2477.89, "word": " the", "probability": 0.90771484375}, {"start": 2477.89, "end": 2478.29, "word": " negative", "probability": 0.91357421875}, {"start": 2478.29, "end": 2479.71, "word": " z.", "probability": 0.469482421875}, {"start": 2480.37, "end": 2480.83, "word": " It's", "probability": 0.964599609375}, {"start": 2480.83, "end": 2481.11, "word": " negative", "probability": 0.69384765625}, {"start": 2481.11, "end": 2481.39, "word": " 1", "probability": 0.88427734375}, {"start": 2481.39, "end": 2482.01, "word": ".15", "probability": 0.8115234375}, {"start": 2482.01, "end": 2483.57, "word": " minus", "probability": 0.75927734375}, {"start": 2483.57, "end": 2483.91, "word": " 1", "probability": 0.97705078125}, {"start": 2483.91, "end": 2484.39, "word": ".2", "probability": 0.703125}, {"start": 2484.39, "end": 2484.55, "word": " up", "probability": 0.8427734375}, {"start": 2484.55, "end": 2484.67, "word": " to", "probability": 0.96484375}, {"start": 2484.67, "end": 2485.01, "word": " 5.", "probability": 0.8046875}, {"start": 2488.15, "end": 2488.83, "word": " So", "probability": 0.95068359375}, {"start": 2488.83, "end": 2489.17, "word": " 0", "probability": 0.576171875}, {"start": 2489.17, "end": 2490.07, "word": ".1056.", "probability": 0.9850260416666666}, {"start": 2493.73, "end": 2494.41, "word": " So", "probability": 0.9560546875}, {"start": 2494.41, "end": 2494.65, "word": " the", "probability": 0.85400390625}, {"start": 2494.65, "end": 2495.11, "word": " answer", "probability": 0.953125}, {"start": 2495.11, "end": 2495.61, "word": " is", "probability": 0.9453125}, {"start": 2495.61, "end": 2496.07, "word": " around", "probability": 0.890625}, {"start": 2496.07, "end": 2496.45, "word": " 10", "probability": 0.94482421875}, {"start": 2496.45, "end": 2496.75, "word": "%", "probability": 0.96337890625}, {"start": 2496.75, "end": 2497.91, "word": " of", "probability": 0.96630859375}, {"start": 2497.91, "end": 2498.09, "word": " the", "probability": 0.9228515625}, {"start": 2498.09, "end": 2499.13, "word": " catfish", "probability": 0.85546875}], "temperature": 1.0}, {"id": 89, "seek": 252390, "start": 2500.28, "end": 2523.9, "text": " will weigh less than 2 pounds. So the answer is 0.1056. Questions? So go back to the PowerPoint presentation we have. The last topic, evaluating normality.", "tokens": [486, 13843, 1570, 813, 568, 8319, 13, 407, 264, 1867, 307, 1958, 13, 3279, 18317, 13, 27738, 30, 407, 352, 646, 281, 264, 25584, 5860, 321, 362, 13, 440, 1036, 4829, 11, 27479, 2026, 1860, 13], "avg_logprob": -0.20682010618416038, "compression_ratio": 1.21875, "no_speech_prob": 0.0, "words": [{"start": 2500.28, "end": 2500.66, "word": " will", "probability": 0.33447265625}, {"start": 2500.66, "end": 2501.06, "word": " weigh", "probability": 0.830078125}, {"start": 2501.06, "end": 2501.56, "word": " less", "probability": 0.93017578125}, {"start": 2501.56, "end": 2502.2, "word": " than", "probability": 0.93701171875}, {"start": 2502.2, "end": 2502.64, "word": " 2", "probability": 0.81689453125}, {"start": 2502.64, "end": 2503.56, "word": " pounds.", "probability": 0.6171875}, {"start": 2503.68, "end": 2503.82, "word": " So", "probability": 0.79150390625}, {"start": 2503.82, "end": 2503.94, "word": " the", "probability": 0.7880859375}, {"start": 2503.94, "end": 2504.16, "word": " answer", "probability": 0.95751953125}, {"start": 2504.16, "end": 2504.48, "word": " is", "probability": 0.9423828125}, {"start": 2504.48, "end": 2505.26, "word": " 0", "probability": 0.8095703125}, {"start": 2505.26, "end": 2506.42, "word": ".1056.", "probability": 0.9775390625}, {"start": 2508.34, "end": 2509.22, "word": " Questions?", "probability": 0.63037109375}, {"start": 2512.78, "end": 2513.66, "word": " So", "probability": 0.86962890625}, {"start": 2513.66, "end": 2513.82, "word": " go", "probability": 0.65380859375}, {"start": 2513.82, "end": 2514.06, "word": " back", "probability": 0.87158203125}, {"start": 2514.06, "end": 2514.22, "word": " to", "probability": 0.9658203125}, {"start": 2514.22, "end": 2514.38, "word": " the", "probability": 0.92236328125}, {"start": 2514.38, "end": 2514.68, "word": " PowerPoint", "probability": 0.61474609375}, {"start": 2514.68, "end": 2515.54, "word": " presentation", "probability": 0.92578125}, {"start": 2515.54, "end": 2515.82, "word": " we", "probability": 0.7236328125}, {"start": 2515.82, "end": 2516.1, "word": " have.", "probability": 0.6591796875}, {"start": 2517.78, "end": 2518.12, "word": " The", "probability": 0.88330078125}, {"start": 2518.12, "end": 2519.12, "word": " last", "probability": 0.8876953125}, {"start": 2519.12, "end": 2519.58, "word": " topic,", "probability": 0.9306640625}, {"start": 2522.56, "end": 2523.14, "word": " evaluating", "probability": 0.88525390625}, {"start": 2523.14, "end": 2523.9, "word": " normality.", "probability": 0.9638671875}], "temperature": 1.0}, {"id": 90, "seek": 255251, "start": 2526.93, "end": 2552.51, "text": " Many times we mentioned something about normality and outliers. For sure, if outliers exist, in this case, the situation is not normal. Now, how can we tell if a data point is an outlier? If you remember, we talked about outliers in Chapter 3. 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And we mentioned that any data point.", "tokens": [3146, 341, 6175, 13, 400, 321, 2835, 300, 604, 1412, 935, 13], "avg_logprob": -0.4825720970447247, "compression_ratio": 0.8833333333333333, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2556.75, "end": 2557.59, "word": " By", "probability": 0.269775390625}, {"start": 2557.59, "end": 2557.87, "word": " this", "probability": 0.546875}, {"start": 2557.87, "end": 2558.27, "word": " score.", "probability": 0.63330078125}, {"start": 2562.65, "end": 2563.49, "word": " And", "probability": 0.892578125}, {"start": 2563.49, "end": 2563.69, "word": " we", "probability": 0.888671875}, {"start": 2563.69, "end": 2564.11, "word": " mentioned", "probability": 0.64306640625}, {"start": 2564.11, "end": 2564.51, "word": " that", "probability": 0.7568359375}, {"start": 2564.51, "end": 2566.71, "word": " any", "probability": 0.66015625}, {"start": 2566.71, "end": 2566.97, "word": " data", "probability": 0.92724609375}, {"start": 2566.97, "end": 2567.39, "word": " point.", "probability": 0.96826171875}], "temperature": 1.0}, {"id": 92, "seek": 260323, "start": 2574.95, "end": 2603.23, "text": " Below minus 3, I mean smaller than minus 3, or above 3, these points are suspected to be outliers. 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So we have lower limit, upper limit. So lower limit.", "tokens": [759, 321, 14722, 264, 3126, 4948, 11, 597, 307, 1249, 16, 3175, 502, 13, 20, 28921, 49, 670, 264, 6597, 4948, 13, 407, 321, 362, 3126, 4948, 11, 6597, 4948, 13, 407, 3126, 4948, 13], "avg_logprob": -0.2486979299121433, "compression_ratio": 1.4505494505494505, "no_speech_prob": 0.0, "words": [{"start": 2605.43, "end": 2605.59, "word": " If", "probability": 0.6318359375}, {"start": 2605.59, "end": 2605.81, "word": " we", "probability": 0.93017578125}, {"start": 2605.81, "end": 2606.25, "word": " compute", "probability": 0.91162109375}, {"start": 2606.25, "end": 2606.45, "word": " the", "probability": 0.9130859375}, {"start": 2606.45, "end": 2606.65, "word": " lower", "probability": 0.88525390625}, {"start": 2606.65, "end": 2607.07, "word": " limit,", "probability": 0.92041015625}, {"start": 2608.91, "end": 2610.25, "word": " which", "probability": 0.94677734375}, {"start": 2610.25, "end": 2610.51, "word": " is", "probability": 0.87939453125}, {"start": 2610.51, "end": 2611.17, "word": " Q1", "probability": 0.712890625}, {"start": 2611.17, "end": 2611.61, "word": " minus", "probability": 0.211669921875}, {"start": 2611.61, "end": 2611.93, "word": " 1", "probability": 0.896484375}, {"start": 2611.93, "end": 2612.55, "word": ".5", "probability": 0.947509765625}, {"start": 2612.55, "end": 2614.89, "word": " IQR", "probability": 0.765625}, {"start": 2614.89, "end": 2616.47, "word": " over", "probability": 0.18212890625}, {"start": 2616.47, "end": 2616.67, "word": " the", "probability": 0.919921875}, {"start": 2616.67, "end": 2616.89, "word": " upper", "probability": 0.83837890625}, {"start": 2616.89, "end": 2617.31, "word": " limit.", "probability": 0.97265625}, {"start": 2620.17, "end": 2620.17, "word": " So", "probability": 0.63427734375}, {"start": 2620.17, "end": 2627.49, "word": " we", "probability": 0.82666015625}, {"start": 2627.49, "end": 2627.61, "word": " have", "probability": 0.9453125}, {"start": 2627.61, "end": 2627.85, "word": " lower", "probability": 0.8359375}, {"start": 2627.85, "end": 2628.21, "word": " limit,", "probability": 0.97021484375}, {"start": 2628.65, "end": 2628.89, "word": " upper", "probability": 0.88330078125}, {"start": 2628.89, "end": 2629.23, "word": " limit.", "probability": 0.96484375}, {"start": 2630.21, "end": 2631.05, "word": " So", "probability": 0.853515625}, {"start": 2631.05, "end": 2631.25, "word": " lower", "probability": 0.7265625}, {"start": 2631.25, "end": 2631.69, "word": " limit.", "probability": 0.96630859375}], "temperature": 1.0}, {"id": 94, "seek": 266504, "start": 2635.4, "end": 2665.04, "text": " And upper limit. Any data point below lower limit or above upper limit is considered to be a type. So therefore, we have two methods to determine or to examine if the observation is enough there or not. One by using this score, straightforward. And the other one, we have to look for", "tokens": [400, 6597, 4948, 13, 2639, 1412, 935, 2507, 3126, 4948, 420, 3673, 6597, 4948, 307, 4888, 281, 312, 257, 2010, 13, 407, 4412, 11, 321, 362, 732, 7150, 281, 6997, 420, 281, 17496, 498, 264, 14816, 307, 1547, 456, 420, 406, 13, 1485, 538, 1228, 341, 6175, 11, 15325, 13, 400, 264, 661, 472, 11, 321, 362, 281, 574, 337], "avg_logprob": -0.2638319574418615, "compression_ratio": 1.569060773480663, "no_speech_prob": 0.0, "words": [{"start": 2635.4, "end": 2635.78, "word": " And", "probability": 0.28857421875}, {"start": 2635.78, "end": 2636.16, "word": " upper", "probability": 0.59228515625}, {"start": 2636.16, "end": 2636.48, "word": " limit.", "probability": 0.94970703125}, {"start": 2639.98, "end": 2640.6, "word": " Any", "probability": 0.92138671875}, {"start": 2640.6, "end": 2641.34, "word": " data", "probability": 0.9404296875}, {"start": 2641.34, "end": 2642.48, "word": " point", "probability": 0.9619140625}, {"start": 2642.48, "end": 2644.36, "word": " below", "probability": 0.67333984375}, {"start": 2644.36, "end": 2644.7, "word": " lower", "probability": 0.45654296875}, {"start": 2644.7, "end": 2645.1, "word": " limit", "probability": 0.966796875}, {"start": 2645.1, "end": 2646.8, "word": " or", "probability": 0.62890625}, {"start": 2646.8, "end": 2647.32, "word": " above", "probability": 0.9599609375}, {"start": 2647.32, "end": 2648.42, "word": " upper", "probability": 0.8447265625}, {"start": 2648.42, "end": 2648.74, "word": " limit", "probability": 0.96533203125}, {"start": 2648.74, "end": 2648.98, "word": " is", "probability": 0.8759765625}, {"start": 2648.98, "end": 2649.3, "word": " considered", "probability": 0.80322265625}, {"start": 2649.3, "end": 2649.56, "word": " to", "probability": 0.96240234375}, {"start": 2649.56, "end": 2649.7, "word": " be", "probability": 0.95068359375}, {"start": 2649.7, "end": 2649.8, "word": " a", "probability": 0.345947265625}, {"start": 2649.8, "end": 2650.18, "word": " type.", "probability": 0.3271484375}, {"start": 2651.34, "end": 2651.96, "word": " So", "probability": 0.826171875}, {"start": 2651.96, "end": 2652.8, "word": " therefore,", "probability": 0.67333984375}, {"start": 2652.94, "end": 2653.08, "word": " we", "probability": 0.94140625}, {"start": 2653.08, "end": 2653.36, "word": " have", "probability": 0.94384765625}, {"start": 2653.36, "end": 2654.22, "word": " two", "probability": 0.93310546875}, {"start": 2654.22, "end": 2654.8, "word": " methods", "probability": 0.9033203125}, {"start": 2654.8, "end": 2656.0, "word": " to", "probability": 0.9384765625}, {"start": 2656.0, "end": 2656.56, "word": " determine", "probability": 0.93212890625}, {"start": 2656.56, "end": 2656.84, "word": " or", "probability": 0.82666015625}, {"start": 2656.84, "end": 2656.96, "word": " to", "probability": 0.9306640625}, {"start": 2656.96, "end": 2657.34, "word": " examine", "probability": 0.955078125}, {"start": 2657.34, "end": 2658.06, "word": " if", "probability": 0.94140625}, {"start": 2658.06, "end": 2658.32, "word": " the", "probability": 0.91357421875}, {"start": 2658.32, "end": 2658.8, "word": " observation", "probability": 0.8701171875}, {"start": 2658.8, "end": 2659.06, "word": " is", "probability": 0.93359375}, {"start": 2659.06, "end": 2659.3, "word": " enough", "probability": 0.5966796875}, {"start": 2659.3, "end": 2659.48, "word": " there", "probability": 0.9013671875}, {"start": 2659.48, "end": 2659.6, "word": " or", "probability": 0.94775390625}, {"start": 2659.6, "end": 2659.86, "word": " not.", "probability": 0.94091796875}, {"start": 2660.22, "end": 2660.52, "word": " One", "probability": 0.9248046875}, {"start": 2660.52, "end": 2660.72, "word": " by", "probability": 0.76123046875}, {"start": 2660.72, "end": 2660.96, "word": " using", "probability": 0.9296875}, {"start": 2660.96, "end": 2661.22, "word": " this", "probability": 0.74853515625}, {"start": 2661.22, "end": 2661.58, "word": " score,", "probability": 0.1737060546875}, {"start": 2661.88, "end": 2662.46, "word": " straightforward.", "probability": 0.84814453125}, {"start": 2663.24, "end": 2663.44, "word": " And", "probability": 0.95703125}, {"start": 2663.44, "end": 2663.56, "word": " the", "probability": 0.7978515625}, {"start": 2663.56, "end": 2663.74, "word": " other", "probability": 0.8876953125}, {"start": 2663.74, "end": 2663.9, "word": " one,", "probability": 0.92724609375}, {"start": 2663.96, "end": 2664.06, "word": " we", "probability": 0.953125}, {"start": 2664.06, "end": 2664.22, "word": " have", "probability": 0.947265625}, {"start": 2664.22, "end": 2664.44, "word": " to", "probability": 0.966796875}, {"start": 2664.44, "end": 2664.66, "word": " look", "probability": 0.96630859375}, {"start": 2664.66, "end": 2665.04, "word": " for", "probability": 0.94970703125}], "temperature": 1.0}, {"id": 95, "seek": 269412, "start": 2665.44, "end": 2694.12, "text": " The lower limit and upper limit. The other method by using software and later on you will take SPSS in order to determine if the data is normally distributed by using something called QQ plot or normal probability plot. So I'm going to skip this part.", "tokens": [440, 3126, 4948, 293, 6597, 4948, 13, 440, 661, 3170, 538, 1228, 4722, 293, 1780, 322, 291, 486, 747, 318, 6273, 50, 294, 1668, 281, 6997, 498, 264, 1412, 307, 5646, 12631, 538, 1228, 746, 1219, 1249, 48, 7542, 420, 2710, 8482, 7542, 13, 407, 286, 478, 516, 281, 10023, 341, 644, 13], "avg_logprob": -0.21874999503294626, "compression_ratio": 1.5089820359281436, "no_speech_prob": 0.0, "words": [{"start": 2665.44, "end": 2665.92, "word": " The", "probability": 0.282958984375}, {"start": 2665.92, "end": 2666.32, "word": " lower", "probability": 0.63671875}, {"start": 2666.32, "end": 2666.64, "word": " limit", "probability": 0.86376953125}, {"start": 2666.64, "end": 2666.92, "word": " and", "probability": 0.86767578125}, {"start": 2666.92, "end": 2667.16, "word": " upper", "probability": 0.72802734375}, {"start": 2667.16, "end": 2667.42, "word": " limit.", "probability": 0.9345703125}, {"start": 2668.96, "end": 2669.18, "word": " The", "probability": 0.8662109375}, {"start": 2669.18, "end": 2669.5, "word": " other", "probability": 0.8828125}, {"start": 2669.5, "end": 2669.92, "word": " method", "probability": 0.9453125}, {"start": 2669.92, "end": 2671.2, "word": " by", "probability": 0.5703125}, {"start": 2671.2, "end": 2671.66, "word": " using", "probability": 0.9365234375}, {"start": 2671.66, "end": 2673.16, "word": " software", "probability": 0.92236328125}, {"start": 2673.16, "end": 2673.56, "word": " and", "probability": 0.6943359375}, {"start": 2673.56, "end": 2673.98, "word": " later", "probability": 0.89794921875}, {"start": 2673.98, "end": 2674.2, "word": " on", "probability": 0.91552734375}, {"start": 2674.2, "end": 2674.3, "word": " you", "probability": 0.85546875}, {"start": 2674.3, "end": 2674.44, "word": " will", "probability": 0.59423828125}, {"start": 2674.44, "end": 2674.74, "word": " take", "probability": 0.8583984375}, {"start": 2674.74, "end": 2676.02, "word": " SPSS", "probability": 0.9119466145833334}, {"start": 2676.02, "end": 2676.78, "word": " in", "probability": 0.79052734375}, {"start": 2676.78, "end": 2676.98, "word": " order", "probability": 0.91943359375}, {"start": 2676.98, "end": 2677.22, "word": " to", "probability": 0.95458984375}, {"start": 2677.22, "end": 2677.68, "word": " determine", "probability": 0.91748046875}, {"start": 2677.68, "end": 2679.2, "word": " if", "probability": 0.89306640625}, {"start": 2679.2, "end": 2679.38, "word": " the", "probability": 0.91357421875}, {"start": 2679.38, "end": 2679.72, "word": " data", "probability": 0.9423828125}, {"start": 2679.72, "end": 2681.18, "word": " is", "probability": 0.9384765625}, {"start": 2681.18, "end": 2681.56, "word": " normally", "probability": 0.8798828125}, {"start": 2681.56, "end": 2682.1, "word": " distributed", "probability": 0.92724609375}, {"start": 2682.1, "end": 2682.44, "word": " by", "probability": 0.857421875}, {"start": 2682.44, "end": 2682.76, "word": " using", "probability": 0.9287109375}, {"start": 2682.76, "end": 2683.1, "word": " something", "probability": 0.8623046875}, {"start": 2683.1, "end": 2683.58, "word": " called", "probability": 0.87939453125}, {"start": 2683.58, "end": 2686.54, "word": " QQ", "probability": 0.740478515625}, {"start": 2686.54, "end": 2686.9, "word": " plot", "probability": 0.53076171875}, {"start": 2686.9, "end": 2688.44, "word": " or", "probability": 0.63623046875}, {"start": 2688.44, "end": 2688.94, "word": " normal", "probability": 0.207275390625}, {"start": 2688.94, "end": 2691.0, "word": " probability", "probability": 0.9462890625}, {"start": 2691.0, "end": 2691.34, "word": " plot.", "probability": 0.84326171875}, {"start": 2691.86, "end": 2692.24, "word": " So", "probability": 0.86279296875}, {"start": 2692.24, "end": 2692.54, "word": " I'm", "probability": 0.77783203125}, {"start": 2692.54, "end": 2692.74, "word": " going", "probability": 0.9462890625}, {"start": 2692.74, "end": 2692.94, "word": " to", "probability": 0.96044921875}, {"start": 2692.94, "end": 2693.22, "word": " skip", "probability": 0.98779296875}, {"start": 2693.22, "end": 2693.66, "word": " this", "probability": 0.94580078125}, {"start": 2693.66, "end": 2694.12, "word": " part.", "probability": 0.8828125}], "temperature": 1.0}, {"id": 96, "seek": 272429, "start": 2695.41, "end": 2724.29, "text": " Because data is taken by using software. But in general, you may look at this graph. 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It's hard to make this graph manual. It's better to use software. But at least if we have this graph, and the points are close to the straight line. I mean, the points are either on the straight line, lies on the straight line, or close it. 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