Page 51 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Note from definition (2.28) that

The standard deviation of a r.v. X, denoted by a,, is the positive square root of Var(X).
Expanding the right-hand side of Eq. (2.28), we can obtain the following relation:

which is a useful formula for determining the variance.

2.7 SOME SPECIAL DISTRIBUTIONS
In this section we present some important special distributions.

A. Bernoulli Distribution:

A r.v. X is called a Bernoulli r.v. with parameter p if its pmf is given by
px(k) = P(X = k) = pk(l - P)'-~ k = 0, 1

where 0 p I 1. By Eq. (2.18), the cdf FX(x) of the Bernoulli r.v. X is given by
x<o

Figure 2-4 illustrates a Bernoulli distribution.

Fig. 2-4 Bernoulli distribution.

The mean and variance of the Bernoulli r.v. X are

A Bernoulli r.v. X is associated with some experiment where an outcome can be classified as
either a "success" or a "failure," and the probability of a success is p and the probability of a failure is
1 - p. Such experiments are often called Bernoulli trials (Prob. 1.61).