Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.5 EXPECTATION FOR DISCRETE SAMPLE SPACES

For discrete random variables with N outcomes, the expectation is given by (5.11). The expectation for a discrete random variable with a countably infinite number of outcomes is derived by considering a sequence of simple random variables as follows. Assume initially that X is nonnegative. It is straightforward to show that a sequence of simple random variables exists that tends to in the limit. For example, assume that and let be the identity mapping so that the outcomes of are nonnegative integers. Define the following sequence of simple random variables:

(5.13)

for . The outcomes of this sequence of simple random variables are 0, {0, 1}, {0, 1, 2}, and so on. Note that by construction, this sequence is nondecreasing for every : . For example, ...