4.1 Definition of expectation
In the previous chapter, we introduced the distribution of a random variable, which gives us full information about the probability that the r.v. will fall into any particular set. For example, we can say how likely it is that the r.v. will exceed 1000, that it will equal 5, and that it will be in the interval [0, 7]. It can be unwieldy to manage so many probabilities though, so often we want just one number summarizing the “average” value of the r.v.
There are several senses in which the word “average” is used, but by far the most commonly used is the mean of an r.v., also known as its expected value. In addition, much of statistics is about understanding variability in the world, so it is ...