Page 103 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES 95

(c) Now
p,(0)py(O) = OS(0.55) = 0.275 # pxy(O, 0) = 0.45
Hence X and Y are not independent.

Consider an experiment of drawing randomly three balls from an urn containing two red, three
white, and four blue balls. Let (X, Y) be a bivariate r.v. where X and Y denote, respectively, the
number of red and white balls chosen.
Find the range of (X, Y).
Find the joint pmf's of (X, Y).
Find the marginaI pmf's of X and Y.
Are X and Y independent?

The range of (X, Y) is given by
Rxr = {(O, 019 (0, I), (0, 2), (0, 3)9 (19 01, (1, 119 (1, 3, (2, O), (2, 1))
The joint pmf's of (X, Y)
pxy(i,j)= P(X=i, Y =j) i=O, 1,2 j=O, 1,2,3
are given as follows:

which are expressed in tabular form as in Table 3.1.

The marginal pmf's of X are obtained from Table 3.1 by computing the row sums, and the marginal
pmf's of Y are obtained by computing the column sums. Thus

Table 3.1 p&i, j)