5. Expectation

The distribution of a random variable \(X\) contains all the information you need to find probabilities of events determined by \(X\). For example, to find the chance that \(X\) is greater than 20, you can identify all the possible values of \(X\) that are greater than 20 and then add up all their chances.

Sometimes, we don’t need all this information. We might just want to know roughly how big \(X\) is. For this we need a rough sense of where the distribution of \(X\) is situated on the number line.

In this chapter we will study a quantity called the expectation of \(X\) that is a measure of the location of the distribution. In data science, the expectation of \(X\) sometimes serves as a guess for how big \(X\) can be. In later chapters we will study how good it is as a guess.