Page 155 - Schaum's Outlines - Probability, Random Variables And Random Processes

FUNCTIONS OF RANDOM VARIABLES, EXPEC'TATION, LIMIT THEOREMS [CHAP. 4

(b) Using.Eq. (4.1 Oj), we have

4.38. Verify Eq. (4.39).
Using Eqs. (3.58) and (3.38), we have

= J:* LUy ZT MX) dx dl. = J:Y[J:fxyix, Y) dx] dy
fxr(x, Y)
-

4.39. LetZ=aX+ bY,whereaandbareconstants.Show that

We verify for the continuous case. The proof for the discrete case is similar.

Note that Eq. (4.107) (the linearity of E) can be easily extended to n r.v.'s:

4.40. Let Y = ax + b.
Find the covariance of X and Y.
Find the correlation coefficient of X and Y.
By Eq. (4.107), we have
E(XY) = E[X(aX + h)] = aE(X2) + bE(X)
E(Y) = E(aX + h) = aE(X) + b
Thus, the covariance of X and Y is [Eq. (3.5111

By Eq. (4.106), we have a, = / a I a,. Thus, the correlation coefficient of X and Y is [Eq. (3.53)J