Page 47 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Fig. 2-2 One random variable associated with coin tossing.

These events have probabilities that are denoted by
P(X = x) = P{C: X(6) = X}
P(X 5 x) = P(6: X(6) 5 x}
P(X > x) = P{C: X(6) > x)
P(x, < X I x,) = P{(: x, < X(C) I x,)
EXAMPLE 2.2 In the experiment of tossing a fair coin three times (Prob. 1.1), the sample space S, consists of
eight equally likely sample points S, = (HHH, . . . , TTT). If X is the r.v. giving the number of heads obtained, find
(a) P(X = 2); (b) P(X < 2).
(a) Let A c S, be the event defined by X = 2. Then, from Prob. 1.1, we have
A = (X = 2) = {C: X(C) = 2) = {HHT, HTH, THH)
Since the sample points are equally likely, we have
P(X = 2) = P(A) = 3
(b) Let B c S, be the event defined by X < 2. Then
B = (X < 2) = {c: X(() < 2) = (HTT, THT, TTH, TTT)
and P(X < 2) = P(B) = 3 = 4

2.3 DISTRIBUTION FUNCTIONS
A. Definition :
The distribution function [or cumulative distributionfunction (cdf)] of X is the function defined by

Most of the information about a random experiment described by the r.v. X is determined by the
behavior of FAX).

B. Properties of FAX) :
Several properties of FX(x) follow directly from its definition (2.4).

2. Fx(xl) I Fx(x,) if x, < x2
3. lim F,(x) = Fx(oo) = 1
x-'m
4. lim FAX) = Fx(- oo) = 0
x-r-m
5. lim FAX) = Fda+) = Fx(a) a+ = lim a + E
x+a+ O<&+O