Page 105 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

By Eq. (3.21), the marginal pmf's of Y are

(c) Now
Px(xi)P~(Yj) = &ixi2yj = PxAxi Yj)
Hence X and Y are independent.

3.16. Consider an experiment of tossing two coins three times. Coin A is fair, but coin B is not fair,
with P(H) = and P(T) = $. Consider a bivariate r.v. (X, Y), where X denotes the number of
heads resulting from coin A and Y denotes the number of heads resulting from coin B.
(a) Find the range of (X, Y).
(b) Find the joint pmf's of (X, Y).
(c) Find P(X = Y), P(X > Y), and P(X + Y 1 4).
(a) The range of (X, Y) is given by
Rxy = {(i, j): i, j = 0, 1, 2, 3)
(b) It is clear that the r.v.'s X and Y are independent, and they are both binomial r.v.'s with parameters (n,
p) = (3, 4) and (n, p) = (3, $), respectively. Thus, by Eq. (2.36), we have

Since X and Y are independent, the joint pmf's of (X, Y) are

which are tabulated in Table 3.2.
(c) From Table 3.2, we have

Table 3.2 pm(i, J)