Question

#2. (24 points) Let X and Y have joint density (a) Find the marginal pdf of Y. Use it to find E(Y) (b) Give an integral expression for P(X + Y < 0.75), but do not evaluate. (c) Give an integral expression for E(XY), but do not evaluate.

Answer 1

- #1 Giren density function of x is fxoa x2 ocrc 3 o;o.w 2.0 @ PCossX1200] = { foras da ج 2.0 c - ای L 4 10.5 (2:0)3-(0.53 27 8-0.125 = - 27 = 7.875 27 ²/P(0.55X420]=0.2117 b) The mean of x is given by - uZE(x) - u= Să fxca, da

wa f'x. ze da aft () **{36-0] rul = 2.25 / The variance of x is given by V(x)= EC4%) -[Ecx)]? where, ECx2) = 5x facanda - ***x da 22 my a da V Il 100 a ll = 3 = 242 = 5.4 ...E(x2) = 514

hence, VCX = 5.4-(2.25) = 5:4-50625 = 0.3375 -- 100%) = 0.3375) -C) The CDf of .x is obtained as. Excala § faltselt Este de 5 stelt aft () Excal=co; aso [ 27 et ; ocaa3 L ' ; 273

graph Yanis of CDF is an d FX (2) that aroo), i aris Let W=3x+2 var(W) = var (3x+2) = g var cx) : v(ax+b)= q?v(8) = gx vac() - 9X0.3375 = 3.0375 ²V(W)=3.0375