Chapter 5Conditional Distribution and Expectation

5.1 Introduction

We have seen that if two random variables are independent, then their joint distribution can be determined from their marginal distribution functions. In the case of dependent random variables, however, the joint distribution can not be determined in this simple fashion. This leads us to the notions of conditional pmf, conditional pdf, and conditional distribution.

Recalling the definition of conditional probability, c05-math-0001, for two events A and B, we can define the conditional probability c05-math-0004 of event A, given that the event c05-math-0006 has occurred, as

whenever c05-math-0008. In Chapter 3 we noted that if X is a continuous random variable, then c05-math-0010 for all x. In this case, definition (5.1) is not satisfactory. On the other hand, if X is a discrete random variable, then definition (5.1) is adequate, as shown in the following definition.

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