A certain small town, whose population consists of 100 families, has 30 families .... A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one these children is 1 if the child is the firstborn, 2 if the child is the secondborn, and 3 if the child is the thirdborn.
a) A random family is chosen (with equal probabilities), and then a random child within that family is chosen (with equal probabilities). Find the PMF, mean, and the variance of the child’s birth rank.
b) A random child is chosen in the town (with equal probabilities). Find the PMF, mean, and variance of the child’s birth rank.

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*For part A I got $P(X = 1) = 37/60, P(X = 2) = 19/60, P(X = 3) = 4/60$, $E(X) = 1(37/60) + 2(19/60) + 3(4/60) = 1.45$, and $\operatorname{Var}(x) = 149/60 - (1.45)^2= 457/1200$

*For part B I got $E(x) = 1(100/190) + 2(70/190) + 3(20/190) = 1.579\dots$ and $\operatorname{Var}(x) = 2.947 -(1.579)^2= 0.454\dots$

*However, I feel like part B is off but I am not sure where, it just seems incorrect. Any help would be greatly appreciated!


No, your calculations are all correct. That's just how the numbers fall here.
