Page 117 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

(a) Are X and Y independent?
(b) Are X and Y correlated?
(a) Setting R = 1 in the results of Prob. 3.21, we obtain

Since fxy(x, y) # fx(x) fy(y); X and Y are not independent.
(b) By Eqs. (3.47~) and (3.47b), the means of X and Y are

since each integrand is an odd function.
Next, by Eq. (3.43),

The integral vanishes because the contributions of the second and the fourth quadrants cancel those of
the first and the third. Hence, E(XY) = E(X)E(Y) = 0 and X and Y are uncorrelated.

CONDITIONAL MEANS AND CONDITIONAL VARIANCES
3.39. Consider the bivariate r.v. (X, Y) of Prob. 3.14 (or Prob. 3.26). Compute the conditional mean
and the conditional variance of Y given xi = 2.

From Prob. 3.26, the conditional pmf pyl,(yj I xi) is
2xi + y,
I
xi)
PY l~(~j - 1, 2; xi = 1, 2
=
yj =
4xi + 3
12)
T~US, PY~X(Y~ 4+yj yj=1,2
=
and by Eqs. (3.55) and (3.56), the conditional mean and the conditional variance of Y given xi = 2 are

(ij - E) (?)
4+yj
= E[(Y - {y I xi = 21 =