ABSTRACT
This chapter extends the concept of conditional probability to random variables. To avoid repeatedly using the term mass/density we will just use joint, conditional and marginal when describing results that hold for both discrete and continuous cases. In general, we can decompose the joint mass/density function of a random vector into a product of conditional mass/density functions. The chapter deals with results that arise when we take expectations with respect to a conditional distribution. One key distinction is that, while unconditional expected values are just numbers, conditional expectation is a random variable. Conditional moments are defined as conditional expectations. Thus, each conditional moment is a function of the random variable we are conditioning on and, as such, is itself a random variable. The conditional moment-generating function can be used to calculate the marginal moment-generating function of Y and the joint moment-generating function of X and Y.
