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

E.8 PMF SUMS AND FUNCTIONAL FORMS

Each probability mass function (pmf) covered in Chapter 3 is either a finite sum or an infinite series whose sum is 1. The elements of a pmf have a functional form as shown in Table E.3 for the various random variables (excluding the discrete uniform pmf), where constants that are independent of x have been dropped. Without these weightings, the functional forms sum to the values in the last column of the table. The first five functions are all based on Bernoulli experiments with probability of success p.

TABLE E.3 Pmf Functional Forms for Discrete Random Variables

It is the functional form that defines the pmf, whereas the inverse of the quantity in the last column of the table is just a weighting included to normalize the function so that the pmf sums to 1. For the Poisson random variable, for example, the functional form in the second column of the table is a series for the exponential function. Thus, the inverse of the sum given by weights the functional form to give a valid pmf. Likewise for the hypergeometric random variable, the functional form must be scaled by to obtain a valid pmf. These pmf sums can be used to find closed-form expressions ...