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

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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:

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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

(5.282) Numbered Display Equation

where the last sum is over a valid pmf and is equal to 1. Substituting yields and .

Example 5.36. The second moment of ...

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