Page 48 - Probability, Random Variables and Random Processes

40 RANDOM VARIABLES [CHAP 2

Property 1 follows because FX(x) is a probability. Property 2 shows that FX(x) is a nondecreasing
function (Prob. 2.5). Properties 3 and 4 follow from Eqs. (1.22) and (1.26):
limP(X<x)= P(X < co) = P(S)= 1
X+oO
lim P(X s x) = P(X s - co) = P(0) = 0
x-'-a,
Property 5 indicates that FX(x) is continuous on the right. This is the consequence of the definition
(2.4).

Table 2.1

%
(TTT)
(TTT, TTH, THT, HTT)
(TTT, TTH, THT, HTT, HHT, HTH, THH)
S
S

EXAMPLE 2.3 Consider the r.v. X defined in Example 2.2. Find and sketch the cdf FX(x) of X.
Table 2.1 gives Fx(x) = P(X I x) for x = - 1, 0, 1, 2, 3, 4. Since the value of X must be an integer, the value of
F,(x) for noninteger values of x must be the same as the value of FX(x) for the nearest smaller integer value of x.
The FX(x) is sketched in Fig. 2-3. Note that F,(x) has jumps at x = 0, 1,2,3, and that at each jump the upper value
is the correct value for FX(x).

-I 0 I 2 3 4
Fig. 2-3

C. Determination of Probabilities from the Distribution Function:
From definition (2.4), we can compute other probabilities, such as P(a < X I b), P(X > a), and
P(X < b) (Prob. 2.6):

P(X < b) = F,(b-) b-= lim b-E
O<E-'O