Page 104 - Probability, Random Variables and Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

(d) Since
PXY(0, 0) = 6 Z PX(O)PY(O) = iE (%I
X and Y are not independent.

3.14. The joint pmf of a bivariate r.v. (X, Y) is given by
k(2xi + yj) xi = 1, 2; yj = 1, 2
otherwise
where k is a constant.
(a) Find the value of k.
(b) Find the marginal pmf's of X and Y.
(c) Are X and Y independent?

Thus, k = &.
(b) By Eq. (3.20), the marginal pmf's of X are

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

(c) Now pAxi)py(yj) # pxY(xi, y,); hence X and Y are not independent.
3.15. The joint pmf of a bivariate r.v. (X, Y) is given by

kxi2yj xi=l,2;yj=l,2,3
otherwise
where k is a constant.
Find the value of k.
Find the marginal pmf's of X and Y.
Are X and Y independent?
By Eq. (3.1 7),

Thus, k = &.
By Eq. (3.20), the marginal pmf's of X are