The distribution of a random variable specifies the possible values and the probability of any event that involves the random variable. The distribution of a random variable contains all the information about its long run behavior. It is also useful to summarize some key features of a distribution.
Chapter 2 introduced how simulation can be used to approximate the long run average value of a random variable. We also saw how variance and standard deviation can be used to summarize the overall degree of variability of a random variable by measuring, roughly, the long run average distance from the mean. We saw a brief introduction to correlation, a number which measures the degree of the relationship between two jointly distributed random variables. Correlation is also based on certain long run averages.
In this chapter we will see how characteristics based on long run averages can be defined as “expected” values. We will see some properties and some strategies for computing expected values. We will also discuss how to interpret expected values, where we’ll see that the “expected” in expected value is, unfortunately, a misnomer (hence the quotes).