Page 149 - Schaum's Outlines - Probability, Random Variables And Random Processes

142 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

Fig. 4-8

4.26. Let X and Y be two r.v.3 with joint pdf f,Ax, y) and joint cdf F&, y). Let W = min(X, Y).
(a) Find the cdf of W.
(b) Find the pdf of W if X and Y are independent.
(a) The region in the xy plane corresponding to the event {min(X, Y) I w) is shown as the shaded area in
Fig. 4-9. Then
P(W I w) = P((X I w) u (Y I w)}
= P(X 5 w) + P(Y I - P((X < w) n (Y 5 w))
w)

(b) If X and Y are independent, then

and differentiating with respect to w gives

Fig. 4-9

4.27. Let X and Y be two r.v.'s with joint pdf f,& y). Let

Find f,,(r, 0) in terms off,&, y).