Introduction to Probability, Second Edition, 2nd Edition

6

Moments

The nth moment of an r.v. X is $E\left(X^n\right)$. In this chapter, we explore how the moments of an r.v. shed light on its distribution. We have already seen that the first two moments are useful since they provide the mean $E\left(X\right)$ and variance $E\left(X^2\right)-{\left(EX\right)}^2$, which are important summaries of the average value of X and how spread out its distribution is. But there is much more to a distribution than its mean and variance. We’ll see that the third and fourth moments tell us about the asymmetry of a distribution and the behavior of the tails or extreme values, two properties that are not captured by the mean and variance. After introducing moments, we’ll discuss the moment generating function (MGF), which not only helps us compute moments but also provides ...