Page 101 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

(b) It is clear that X and Y are independent and
P(X = 0) = P(X = 1) = 4 = q
P(Y=O)=P(Y= I)=$=$
Thus pxy(i,j)=P(X=i, Y=j)=P(X=i)P(Y =j)=$ i,j=O, 1

3.12. Consider the binary communication channel shown in Fig. 3-4 (Prob. 1.52). Let (X, Y) be a
bivariate r.v., where X is the input to the channel and Y is the output of the channel. Let
P(X = 0) = 0.5, P(Y = 11 X = 0) = 0.1, and P(Y = 0 1 X = 1) = 0.2.
Find the joint pmf's of (X, Y).
Find the marginal pmf's of X and Y.
Are X and Y independent?
From the results of Prob. 1.52, we found that

Then by Eq. (1.41), we obtain

Hence, the joint pmf's of (X, Y) are

By Eq. (3.20), the marginal pmf's of X are
px(0) = pxu(O, yj) = 0.45 + 0.05 = 0.5
YJ
px(l) = C pxr(l, yj) = 0.1 + 0.4 = 0.5
YJ
By Eq. (3.21), the marginal pmf's of Y are

Fig. 3-4 Binary communication channel.