Page 120 - Probability, Random Variables and Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

Next, we treat zero as "success." If Y denotes the number of successes, then Y is a binomial r.v. with
parameters (n, p) = (4, e-2). Thus, the probability that exactly one of the X,'s equals zero is given by [Eq.
(2.W

3.43. Let (X, Y, Z) be a trivariate r.v., where X, Y, and Z are independent uniform r.v.'s over (0, 1).
Compute P(Z 2 X Y).
Since X, Y, Z are independent and uniformly distributed over (0, I), we have

Then

3
(1
(
= 1 I xy) dy dx = [ - 2) dx =I
3.44. Let (X, Y, Z) be a trivariate r.v. with joint pdf
{;-W+b+W x>O,Y>O,Z>O
fxuz(x, Y, 4 =
otherwise
where a, b, c > 0 and k are constants.
(a) Determine the value of k.
(b) Find the marginal joint pdf of X and Y.
(c) Find the marginal pdf of X.
(d) Are X, Y, and Z independent?

Thus k = abc.
(b) By Eq. (3.77), the marginal joint pdf of X and Y is

(c) By Eq. (3.78), the marginal pdf of X is

(d) Similarly, we obtain