Question

please make sure to give the answer for the second question.

#2. (24 points) Let X and Y have joint density f(x,y) = 60xy2, 0<x<1, 0<y<1, x+y<1. (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

Sal. @ The joint density of x and y fays = Gonya, oraal, ocycl, xeyal Marginal pdf of Y fy(y) = f(x,y) dx l Support of flans - boy? x ( 27 - 30y? (1-73², ocyal €(t) = soy full dy = f & Boy? (-y}] dy = 30. J'y 3 (1+y"-2y) dy = 30. [ + y 245] = 30x40 = ť = 0.5 © P&#92075) Santa dojde 0.75 aty=0.75 (x + y < 11 ( My (0,75) rity=1 © Exy) =s say ffanys dy da رهر6) (۱۰) (219) nir l-a Since Median is the point which divides the entire distribution in two equal past, so if M is the median of y then sa foldy = S froidy = Ź S 30y+(1+y)? dy = 0.5 5 6M5-15m4+10m3 - 0.5= 0 M=0.5 So mean = median Since fly)=304² (1-4)² = 30(y"_2y3 + y²) f'(y) = 30 (y3 – by²+2y) = f'() =0 gives y=0, 1, ½ fl(y) = 30 (1242-12y+2) at y=% f (y) Lo so mode = 0.5 Clearly Mean = Median = Mode - So Y is a symmetrical distribution (zie Not a left skew of right skew )