Question

(50 points) Suppose that the joint p.d.f. of X and Y is as follows: for x 2 0, y 2 0, and x + y <1 elsewhere 2. 24xy f(x)0 a) Determine the value of P(X < Y). b) Determine the marginal p.d.f.'s for Xand Y c) Find P(X> 0.5) d) Determine the conditional p.d.f. of X|Y = 0.5 e) Find P(X> 0.5|Y 0.5) f) Find P(X> 0.5|Y> 0.5) g) Find Cov (X, Y)

Answer 1

2) pdf of x The and joint is y xt' iv(x,y)=(24ny, no, yoo, n+yal 0.w . ylu,y) dyda nly 24 ny dydn

= | 10 + 9 X > » у = х to, o o ,) А - { (, y) : o4 ,, х : у 4 | - у

24 d . M 2 - 2 3 x + x - - 2 V2 a {1-2n] da = 12 1/2 dn - n du - 12 12

3 -2 24 b) pat of The marginal x is fx (2) - Sexy(x,y) dy , XER 1-n 24 nydy, oral 7 0.w.

= ( 24 n I ydy olull ..w 1 x 2 - olac =(24* [ ut ] ,ocx , 0.w. =124n. (1-1) Ocncl , 0.W. ( 12n Ocull 0.W .

eat of y is The marginal fu ulx,y) dr, YER 1-Y I 24nydn, olyal O , O.W. l-y ndn, ocycl =( 244 en 0.w. 21- y = ( 244 [ 7 ] , oryal o.w .

( 244. (1-4)", orgel ocyal 0 , 0.W. ocycl = (12yll-yg- ocycl 0.W . c) P ( x > 0.5) 1- Pl X=0.5) 0.5 (a)da 0.5 aliongran 0.5 = 1- 12 a 11-20 + x²) dx

0.5 I 2 In 1 - 121 olo ndu - 2 0.5 2 0.5 = 1- 123 0.5 21-12{ (0,5). 2x(0.5) 4

1-12{ $ 1232) I 12 x 12x 96 The pdf conditional X given y=y of is Excy (nly) fx, y (x, y) - fyly) NER

24 ny , Oluli-y 12 y(1-4)2 i o .w . 20 oln Lly 1 0 . W . . pdf The conditional of XlY=0.5 is txiy 12/0.5) 2 n ocul I-0.5 11-0.5) ² 0 , 0.w. on 0.5 o.w.