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

3.6 MIXED DISTRIBUTIONS

Starting with the mixed probability in (3.22), we would like to write an expression for a mixed cdf: one that has discontinuities with absolutely continuous parts in between the discontinuities, as depicted by the example in Figure 3.11(c). Let event E in (3.22) be . Then we can write the cdf as follows:

(3.34)

This decomposition implies a conditioning on the values of where the distribution has discontinuities. The subscript s means the following:

(3.35)

where S is the set of values at the discontinuities (the singular discrete components of the cdf). Likewise

(3.36)

where Sc is the support of X excluding the discontinuities. (Actually, the elements of S would not be “removed” from the support of the absolutely continuous part. Since those elements have zero probability in the continuous part, they can be included in (3.36) without affecting the probability). From these definitions, (3.34) has been written in terms of conditional probabilities with conditioning ...