5.21 HIGHER ORDER MOMENTS AND CUMULANTS
In this section, we extend the definitions of mean and variance, which are first- and second-order moments, to higher order moments.
Definition: Moment The nth moment of random variable X is
This type of expectation where the mean is not subtracted is also referred to as a raw moment (noncentral).
Example 5.35. For the Poisson random variable with parameter α, the second moment can be derived by first finding a closed form for the following expression:
which allows us to cancel terms in the denominator and change the lower limit to n = 2. We can factor and change variables in the sum to m = n−2, yielding
where the last sum is over a valid pmf and is equal to 1. Substituting yields and .
Example 5.36. The second moment of ...