Question

PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).

Answer 1

Answer - joint pdf is f(x,y)= txe e, olyuro x=y 9) Pexsy) = ( (*Jlx.y) dy dek lo e adx $18=1 6) marginal of 7 f(Y)= (o trasy) da olycoo 35 xedda ورسم - (س ) - = [ yey_ (0-2)]

419) && (utlery osyco . conditional disth of a given ty f[xly=y) = f(x,y) tly) & £xe-x lythe-y g(8y=y)= x é (89); x>0 How marginal of x is (a) se trendy; ocx = $x-e-2 290 since f(x)y=4) & f(x) Hence x and y are not indepen- c) E(XIysy)= Soox. f(x)y=y)dx dent.

xpx* xp & (25) x ola xp EP y de he dado GP (h) the x ol = (2x)} pro E = (y+ ] P3 FP RP (Ith) ST E KP (6) +6 . (6) XP-2-2,**.*a apral fx al = (x)} (p 100 xe tax

B 4x3+2 = 6 424 Coulx, y) = Elxy) - E(X) Ely) = 6- 3x3