Page 100 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIA.BLES

By De Morgan's law (1.1 5), we have
-
--
(X > x) n (Y > y) = (X > x) u (Y > y) = (X I x) u (Y 5 y)
Then by Eq. (1 D),
P[(X > X) n (Y > y)] = P(X I x) + P(Y 5; y) - P(X s x, Y s y)
= Fx(4 + FAY) - F,,(x, Y)
= (1 - e-"") + (1 - e-fly) - (1 - e-ax)(l - e-fly)
= 1 - e-aXe-By

Finally, by Eq. (1.25), we obtain
P(X > x, Y > y) = 1 - P[(X > x) n (Y > y)] = e-""e-DY

3.9. The joint cdf of a bivariate r.v. (X, Y) is given by

Find the marginal cdf's of X and Y.
Find the conditions on p,, p, , and p, for which X and Y are independent.
By Eq. (3.13), the marginal cdf of X is given by

By Eq. (3.1 4), the marginal cdf of Y is given by

For X and Y to be independent, by Eq. (3.4), we must have FXy(x, y) = FX(x)Fr(y). Thus, for 0 x < a,
0 I y < b, we must have p, = p, p, for X and Y to be independent.

DISCRETE BIVARIATE RANDOM VARIABLESJOINT PROBABILITY MASS
FUNCTIONS
3.10. Verify Eq. (3.22).
If X and Y are independent, then by Eq. (1.46),
pxdxi, yj) = P(X = Xi ,Y = yi) = P(X = xi)P(Y = yj) = P~(X~)P~(Y~)

3.11. Two fair dice are thrown. Consider a bivariate r.v. (X, Y). Let X = 0 or 1 according to whether
the first die shows an even number or an odd number of dots. Similarly, let Y = 0 or 1 according
to the second die.
(a) Find the range Rxy of (X, Y).
(b) Find the joint pmf's of (X, Y).