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

4.3 JOINT AND MARGINAL DISTRIBUTIONS

We are interested in a joint probabilistic description for multiple random variables so that their relationship can be quantified. This is important in many applications such as estimation (see Chapters 9 and 11) where properties of a random variable or a random process can be estimated or predicted from observations of another random quantity. The joint pdf of two random variables X and Y is derived in a manner similar to that used for a single random variable in Chapter 3. We defer the description of mappings from an abstract probability space to multiple random variables until the section on random vectors. Here, we simply give the joint pdf as a definition.

Definition: Joint Probability Density Function The joint pdf of random variables X and Y is denoted by fX,Y(x, y) from which probabilities are computed as follows:

(4.18)

This probability measure is defined on the measurable space and is absolutely continuous with respect to Lebesgue measure on .

The probability above is obtained by integrating the joint pdf over the Cartesian product , which ...