Page 80 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

Substituting B = (a < X 5 b) in Eq. (2.59), we have
P((X I x) n (a < X I b)}
FX(x(a<XIb)=P(X<xlac~Ib)=
P(a < X I b)

Now

Hence,
xla

By Eq. (2.63), the conditional pdf of X given a < X I b is obtained by differentiating Eq. (2.104) with
respect to x. Thus,

Fx(x

2.51. Recall the parachuting sky diver problem (Prob. 2.48). Find the probability of the sky diver
landing within a 10-m radius from the center of the target area given that the landing is within
50 m from the center of the target area.
From Eq. (2.75) (Prob. 2.23) with a2 = 100, we have

Setting x = 10 and b = 50 and a = - cc in Eq. (2.104), we obtain

2.52. Let X = N(0; 02). Find E(X I X > 0) and Var(X 1 X > O),
From Eq. (2.52), the pdf of X = N(0; a2) is

Then by Eq. (2.105),

Hence,