Introduction to Probability and Stochastic Processes with Applications

CHAPTER 6

CONDITIONAL EXPECTATION

One of the most important and useful concepts of probability theory is the conditional expected value. The reason for it is twofold: in the first place, in practice usually it is interesting to calculate probabilities and expected values when some partial information is already known. On the other hand, when one wants to find a probability or an expected value, many times it is convenient to condition first with respect to an appropriate random variable.

6.1   CONDITIONAL DISTRIBUTION

The relationship between two random variables can be seen by finding the conditional distribution of one of them given the value of the other. In Chapter 1, we defined the conditional probability of an event A given another event B as:

It is natural, then, to have the following definition:

Definition 6.1  (Conditional Probability Mass Function) Let X and Y be two discrete random variables. The conditional probability mass functio of X given Y = y is defined as

for all y for which P (Y = y) > 0.

Definition 6.2  (Conditional Distribution Function) The conditional distribution function of X given Y = y is defined as

for all y for which P (Y = y) > 0.

Definition 6.3  (Conditional ...