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

4.5 CONDITIONAL DISTRIBUTION

The conditional distribution of random variable X is a probability measure on the measurable space conditioned on an event or another random variable. This conditioning, in effect, provides information about the experiment and modifies the sample space. The notion of conditioning is the same as that discussed in Chapter 2 in connection with the probabilities of events; it is extended here to the cdf and pdf of a random variable.

Definition: Conditional Distribution Function The cdf of random variable X conditioned on event E is

(4.32)

where it is assumed that .

Generally, there is no reason to condition on an event with zero probability. Observe that FX|E(x|E) is a valid cdf (probability measure) with respect to random variable X: it has the properties described in Chapter 3. It is not a probability measure with respect to the conditioning event E. The conditional cdf is depicted in Figure 4.10 where event A in the abstract sample space Ω corresponds to for random variable X. Since we are interested in random variables, event E might represent for some ...