Page 112 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 105

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3.27. Find the conditional pmf's pylx(yj I xi) and pXIy(xi yj) for the bivariate r.v. (X, Y) of Prob. 3.15.
From the results of Prob. 3.15, we have
xi = 1, 2; yj = 1, 2, 3
otherwise
px(xJ = +xi2 Xi = 1, 2
pAyj)=iyj Yj=1,2,3
Thus, by Eqs. (3.33) and (3.34),

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Note that PYlAyj xi) = py(yj) and pXl& 1 yj) = pX(xi), as must be the case since X and Y are independent,
as shown in Prob. 3.15.

3.28. Consider the bivariate r.v. (X, Y) of Prob. 3.1 7.
(a) Find the conditional pdf's f&(y I x) and fxl,(x 1 y).
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(b) Find P(0 < Y < ) X = 1).
(a) From the results of Prob. 3.17, we have
o<x<2,o<y<2
otherwise
fx(x) = t(x + 1) 0 < x < 2
fdy) = b(y + 1) 0 < Y < 2
Thus, by Eqs. (3.38) and (3.39),

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

3.29. Find the conditional pdf's fy lx(y x) and fxl ,(x ( y) for the bivariate r.v. (X, Y) of Prob. 3.18.
(
From the results of Prob. 3.18, we have
O<x<l,O<y<l
otherwise
fdx) = 2x 0 < x < 1
fv(y) = 2~ 0 < Y < 1
Thus, by Eqs. (3.38) and (3.39),