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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.