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

5.6 EXPECTATION FOR CONTINUOUS SAMPLE SPACES

The expectation of a continuous random variable can also be derived using a sequence of simple random variables. This is done by approximating the continuous random variable with a series of staircase mappings (as is done for a continuous function and the Lebesgue integral in Appendix D). Assume that in (5.1) is nonnegative and define the following simple random variable:

(5.32)

for . This is a countable random variable that approximates the original uncountable random variable X. Note that by construction, the are nondecreasing for every : . The first simple random variable is shown in Figure 5.3. The representation above is similar to quantization ...