# Discrete Random Variables

## Mean or Expected Value and Standard Deviation

The expected value of a discrete random variable  X, symbolized as E(X), is often referred to as the long-term average or  (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average.
As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers.
The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens.
In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency.
Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as 
 
   and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance 
 
   of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance 
 
  . The formulas are given as below.
Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce rounding errors. For some probability distributions, there are shortcut formulas for calculating μ and σ.
Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions.
A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions.

### References

Florida State University. (n.d.). Class catalogue at the Florida State University. Retrieved from https://apps.oti.fsu.edu/RegistrarCourseLookup/SearchFormLegacy
World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction

### Chapter Review

The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes.

### Formula Review

Mean or Expected Value: 
 
Standard Deviation: 
 
Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:
Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:
1. Let X = the number of years a student will study ballet with the teacher.
2. Let P(x) = the probability that a student will study ballet x years.

### HOMEWORK
