Question

3. (50 pts) Let (X, Y) have joint pdf given by c, y x, 0 < x < 1, f(x, y) 0, o.w., (a) Find the constant c. (b) Find fx(x) and fy (y) (c) For 0 < 1, find fyx=x(y) and pyjx=x and oy Y|X=x (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why.

Answer 1

Saya tem, y)C., 191<x, < 0 , a) ; -* <y <x tind c = ? Il fim,y) dy dre= Rr Ry J I c. dy da a l a O -X Sc.(97%dla 21 , s'cc?a) da 21 = 20 (13! = 1 b) fxlay= 5 fixiyody os r. dy = (4- 2n, ochel forme 2n, ornell

Rn Э, 19): 2 ім. 9) 4. -х<< x - - xey [ 94 x >- чек 3,1» | as » (2, +9 , -1<sco f19) J'a» , , , 1-у, оқу <i (9) = 2 г ) +9 , -149 46 К 1-2, o <92) Со , «ә © F. . . . . эки <1 | Ачу. E(9)+) - 9 +1 2+) 44 - 1 (+2 + до 2 (99 2 - 4 + 4 го | Ачук **, Е (**) : у 3, , + Дуі 25" } (+5° - h! (9); E (97%) - t + t = а.н

Pulpen = E(YYxou) - (ElYjxn)]2 2 -(0)2 = x Elmy)= da basta y de day $ say.dusdy a conside 1 2 3 / (x?x23 cm 0 - 3 Elny) 20 EL 9) = {20149) dy + ! 'scr-y.dly (473) +(2-922'--(+3)+(1-1) 2 y31 ż+ 3/6 Ely) - 0 Covin,y)= Elmy) - Elm) Ely 0 - Elm) 0 =0 a cor (X,Y)=0 Ø tim,y)=1 ounci, acych fluva 2n, 0CNC fly) = { 'ty, -15y<o licy ocy Since . f(4,4) # fem fcy). - So X and Y are not independent