Page 119 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 111

3.40. Let (X, Y) be the bivariate r.v. of Prob. 3.20 (or Prob. 3.30). Compute the conditional means
E(Y I x) and E(X I y).
From Prob. 3.30,

By Eq. (3.58), the conditional mean of Y, given X = x, is

Similarly, the conditional mean of X, given Y = y, is

Note that E( Y I x) is a function of x only and E(X I y) is a function of y only.

3.41. Let (X, Y) be the bivariate r.v. of Prob. 3.20 (or Prob. 3.30). Compute the conditional variances
Var(Y I x) and Var(X I y).
Using the results of Prob. 3.40 and Eq. (3.59), the conditional variance of Y, given X = x, is

Var(YIx) = E{CY - E(Y1x)I2 1x1 =

Similarly, the conditional variance of X, given Y = y, is

N-DIMENSIONAL RANDOM VECTORS
3.42. Let (X,, X,, X,, X,) be a four-dimensional random vector, where X, (k = 1, 2, 3, 4) are inde-
pendent Poisson r.v.'s with parameter 2.
(a) Find P(X, = 1, X2 = 3,X3 = 2,X4 = 1).
(b) Find the probability that exactly one of the X,'s equals zero.
(a) By Eq. (2.40), the pmf of X, is

Since the Xis are independent, by Eq. (3.80),
P(X, = 1, x2 = 3, x3 = 2, x4 = 1) = px,(1)px2(3)~x3(2)~x4(1)

(b) First, we find the probability that X, = 0, k = 1, 2,3,4. From Eq. (3.98),
~(x,=O)=e-~ k=1,2,3,4