Page 52 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

B. Binomial Distribution:
A r.v. X is called a binomial r.v. with parameters (n, p) if its pmf is given by

where 0 5 p 5 1 and
n !
(;) = k!(,, - k)!
which is known as the binomial coefficient. The corresponding cdf of X is

Figure 2-5 illustrates the binomial distribution for n = 6 and p = 0.6.

(a (h)
Fig. 2-5 Binomial distribution with n = 6, p = 0.6.

The mean and variance of the binomial r.v. X are (Prob. 2.28)

A binomial r.v. X is associated with some experiments in which n independent Bernoulli trials are
performed and X represents the number of successes that occur in the n trials. Note that a Bernoulli
r.v. is just a binomial r.v. with parameters (1, p).

C. Poisson Distribution:
A r.v. X is called a Poisson r.v. with parameter A (> 0) if its pmf is given by

The corresponding cdf of X is

Figure 2-6 illustrates the Poisson distribution for A = 3.