Page 48 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 21 RANDOM VARIABLES

2.4 DISCRETE RANDOM VARIABLES AND PROBABILITY MASS FUNCTIONS
A. Definition :
Let X be a r.v. with cdf FX(x). If FX(x) changes values only in jumps (at most a countable number
of them) and is constant between jumps-that is, FX(x) is a staircase function (see Fig. 2-3)-- then X
is called a discrete random variable. Alternatively, X is a discrete r.v. only if its range contains a finite
or countably infinite number of points. The r.v. X in Example 2.3 is an example of a discrete r.v.

B. Probability Mass Functions:
Suppose that the jumps in FX(x) of a discrete r.v. X occur at the points x,, x,, . . . , where the
sequence may be either finite or countably infinite, and we assume xi < xj if i < j.
Then FX(xi) - FX(xi- ,) = P(X 5 xi) - P(X I xi- ,) = P(X = xi) (2.1 3)
Let px(x) = P(X = x) (2.1 4)
The function px(x) is called the probability mass function (pmf) of the discrete r.v. X.

Properties of pdx) :

The cdf FX(x) of a discrete r.v. X can be obtained by

2.5 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DENSITY FUNCTIONS
A. Definition:

Let X be a r.v. with cdf FX(x). If FX(x) is continuous and. also has a derivative dFx(x)/dx which
exists everywhere except at possibly a finite number of points and is piecewise continuous, then X is
called a continuous random variable. Alternatively, X is a continuous r.v. only if its range contains an
interval (either finite or infinite) of real numbers. Thus, if X is a. continuous r.v., then (Prob. 2.18)

Note that this is an example of an event with probability 0 that is not necessarily the impossible event
0.
In most applications, the r.v. is either discrete or continuous. But if the cdf FX(x) of a r.v. X
possesses features of both discrete and continuous r.v.'s, then the r.v. X is called the mixed r.v. (Prob.
2.10).

B. Probability Density Functions:

Let

The function fx(x) is called the probability density function (pdf) of the continuous r.v. X.