Page 56 - Probability, Random Variables and Random Processes

48 RANDOM VARIABLES [CHAP 2

2.8 CONDITIONAL DISTRIBUTIONS
In Sec. 1.6 the conditional probability of an event A given event B is defined as

The conditional cdf FX(x ( B) of a r.v. X given event B is defined by

The conditional cdf F,(x 1 B) has the same properties as FX(x). (See Prob. 1.37 and Sec. 2.3.) In
particular,
1
F,(-coIB)=O FX(m B) = 1 (2.60)
P(a < X I b I B) = Fx(b I B) - Fx(a I B) (2.61)
If X is a discrete r.v., then the conditional pmf p,(xk I B) is defined by

If X is a continuous r.v., then the conditional pdf fx(x 1 B) is defined by

Solved Problems

RANDOM VARIABLES
2.1. Consider the experiment of throwing a fair die. Let X be the r.v. which assigns 1 if the number
that appears is even and 0 if the number that appears is odd.
(a) What is the range of X?
(b) Find P(X = 1) and P(X = 0).
The sample space S on which X is defined consists of 6 points which are equally likely:
S = (1, 2, 3, 4, 5, 6)
(a) The range of X is R, = (0, 1).
(b) (X = 1) = (2, 4, 6). Thus, P(X = 1) = 2 = +. Similarly, (X = 0) = (1, 3,5), and P(X = 0) = 3.

2.2. Consider the experiment of tossing a coin three times (Prob. 1.1). Let X be the r.v. giving the
number of heads obtained. We assume that the tosses are independent and the probability of a
head is p.
(a) What is the range of X ?
(b) Find the probabilities P(X = 0), P(X = I), P(X = 2), and P(X = 3).
The sample space S on which X is defined consists of eight sample points (Prob. 1.1):
S= {HHH, HHT, ..., TTT)
(a) The range of X is R, = (0, 1, 2, 3).