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

3.9 DISCRETE RANDOM VARIABLES

As discussed previously, the outcomes of a discrete random variable can be expressed using either the Kronecker delta function or the Dirac delta function, resulting in a pmf or pdf, respectively. Let the outcomes for discrete random variable X be with nonzero probabilities P(X = x). The pmf is written as

(3.240)

and the pdf is

(3.241)

Although the sums above are infinite, only the term for n = x is nonzero because of the delta functions. The indicator function is not needed to specify the support for X (as was used for continuous random variables) because it is evident from the sums above. This is also the case for the cdf where the unit-step function u(x) is used, and which is obtained by integrating the pdf:

(3.242)

In order to have a consistent representation for all random variables (including those with mixed distributions), the pdf representation is generally used in this section. This allows us to use integration later when computing various quantities, such as the mean described in Chapter 5:

(3.243)

where the sifting property of the Dirac ...