Page 125 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

K-l =

Then we can write

Substituting Eqs. (3.1 16) and (3.1 18) into Eq. (3.92), we obtain

Now Eq. (3.1 19) can be rewritten as

where

Thus we conclude that XI, X,, . . . , Xn are independent.

Supplementary Problems

3.55. Consider an experiment of tossing a fair coin three times. Let (X, Y) be a bivariate r.v., where X denotes the
number of heads on the first two tosses and Y denotes the number of heads on the third toss.
(a) Find the range of X.
(b) Find the range of Y.
(c) Find the range of (X, Y).
(d) Find(i)P(X 12, Y I l);(ii)P(X I 1, Y I l);and(iii)P(X 10, Y 10).

3.56. Let FXy(x, y) be a joint cdf of a bivariate r.v. (X, Y). Show that
P(X > a, Y > c) = 1 - Fx(a) - F,(c) + Fx,(a, c)
where F,(x) and F,(y) are marginal cdf7s of X and Y, respectively.
Hint: Set x, = a, y, = c, and x, = y, = co in Eq. (3.95) and use Eqs. (3.1 3) and (3.14).