November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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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
(5.280) 
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:
(5.281) 
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) 
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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