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

5.12 CONDITIONAL EXPECTATION

The conditional probability of events A and B in the abstract probability space was introduced in Chapter 2. There we discussed how conditioning on event B reduces the sample space for event A such that we can write

(5.112)

assuming . In Chapter 3, we considered conditional pdfs which have a form analogous to (5.112):

(5.113)

and likewise for conditional pmfs. The conditional expectation, which is also called the conditional mean, follows from these expressions.

Definition: Conditional Expectation The conditional expectation of random variable X given that random variable Y takes on the value y is

(5.114)

Observe that is a nonrandom function of y, such that we can write . The conditional expectation given random variable Y is itself a random variable because ...