TABLE OF CONTENTS The main focus of this chapter is the study of pairs of continuous random variables that are not independent. In particular, conditional probability and conditional expectation along with corresponding laws of total probability and substitution are studied. These tools are used to compute probabilities involving the output of systems with two (and sometimes three or more) random inputs.
Consider the following functions of two random variables X and Y,
For example, in a telephone channel the signal X is corrupted by additive noise Y. In a wireless channel, the signal X is corrupted by fading (multiplicative noise). If X and Y are the traffic rates at two different routers of an Internet service provider, it is desirable to have these rates less than the router capacity, say u; i.e., we want max( X, Y) ? u. If X and Y are sensor voltages, we may want to trigger an alarm if at least one of the sensor voltages falls below a threshold v; e.g., if min( X, Y) ? v. We now show that the cdfs of these four functions of X and Y can be expressed in the form P(( X, Y) ? A) for various sets1 A ? IR 2. We then argue that such probabilities can be computed in terms of the joint cumulative distribution function to be...
