Page 111 - Schaum's Outlines - Probability, Random Variables And Random Processes

104 MULTIPLE RANDOM VARIABLES [CHAP 3

012, and the possible values of O are between 0 and 42. Thus, the joint pdf of (X, O) is

(0 otherwise
From Fig. 3-10, we see that the condition for the needle to intersect a line is X < L/2 cos 8. Thus, the
probability that the needle will intersect a line is

CONDITIONAL DISTRIBUTIONS
3.25. Verify Eqs. (3.36) and (3.41).
(a) By Eqs. (3.33) and (3.20),

(b) Similarly, by Eqs. (3.38) and (3.30),

3.26. Consider the bivariate r.v. (X, Y) of Prob. 3.14.

(
(a) Find the conditional pmf's fiIx(yj I xi) and pXIY(xi yj).
(b) Find P(Y = 2 1 X = 2) and P(X = 2 1 Y = 2).
(a) From the results of Prob. 3.14, we have
&(2xi + yj) Xi = 1, 2; yj = 1, 2
otherwise
px(xi) = &4xi + 3) xi = 1, 2
P~Y~) Ad2yj + 6) Yj = 17 2
=
Thus, by Eqs. (3.33) and (3.34),

(b) Using the results of part (a), we obtain