Question

55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d) Find the probability P(|X<Y).

Answer 1

Host Ket X and Y be continuous random variables 2 jointly with joint density function f(ny) = be by -a< x < 2a otherwise. eoeyeon 0 xa la こ Given, E[xy] = 1 1 = 1 » Il my fizylite dy & t- you yooo o's be 3y da my do dy yoors-a Mias => 6 of ye. Bylfador dy 47ao * b y ezy a yeo 6. x=2a E I 6 a x = 2a 2 2 x = -a G

=> b. Tye sy (a) & (-a) ² dy z D. G ਨੂੰ 20 : 6 ੪ -31 (4a" 3y dy = 1 ਕੀ bx3c 2 -3 ਤੇ ਹੈ ? ਨੂੰ Let % 2 ਹੈ > 0 ) · ਨੂੰ ਚੀਨ ੨ 30 30 4 > X 36 X ਨੂੰ 4 from gamme function. ?) za ez dz 3a 18 dazla ?) 37 ਤ ਨੂੰ ?) ab UU

Since, the (2) flary) is valid joint joint paf Il flavy) dody I 2 y y20 222a be-3y da dady as ?о х=-а yo x=2a 2) be. By (1 da) dy z z-a za y20 you b. 8-38 [ yed dy 1 ya 2) b b {20-c-a) -34 e -3 20 3ab le al -3 of I દ) gab lo :) ab af

a²b Thus, zf and ab a) lab)xb af =) 1 x b ²1 > b z 1 ( طه ) . а 2 1 Hence, azi bes Ans: Thus, the joint bat of X and Y is green finy) ze By -sexa 2 otherwise oxyco conditional pelf of X green Y where exyco The is Х Y f(0,4) fly Ne2 ply) x=2 e-3y da z 2 2 (fex,y) da fesy e 3y [x]re in Lun20 Lulu'20

Na2 2 fly) e-By [ 1 z 1 2 2 e-38 2-(-1)] e-3y (20) 3e-34 z where of you Thus, the conditional pdf of * giuen y is X 2 fly) e-37 3e-3y N 1 where oxyco and -lehc2 3 Now , E[x]EG] cou (x,y) 2 Elxy] E[X4) 2 (given) 6 Ery] fyfly dy & J y. 30-34 dy Zdz y e o het 7 = 34 ) dz=3dy as yso and a asys 2=0 3 z>

4 2 1.D Y O I Q -3 Z 9 116) 3. XL 3 et They cou (x,y) 2 E[x4] - E[x]E[Y] Liet 3 2 z te の Therefore, zo X and Y are couvriance is o independent and their x=1 PC IXC) region the X=-11 X=2 6 covered by Y=0X .K x 2 and 1XIKY OCY coo

X-੪ ॥2 7 ( 1x1 (1) XL700 dat =-}\r\ Al flm y ely de 1 flory) dy dan %= 0 ਨੂੰ 2 ਹੈ (fe-3ydy) cloc X Yਵ0 =2 e 3 dy) dx et :- 2z . - 40 2 p>21 72 40 da at | 2 ਤੋਂ ਹੈਂ, ? | - 2 20 2 2 3x dat e - e 32 da -3 - -3 ਕ + ) (0 + ) 2 - | 2 s da & da 2 - 1 ? : {z 3 2 0 2 ਪ ਨੇ e - 3 - 3 1 3 3 9 ਟ - Neo T po 3 ੭ - ਰ N 3 3 3 • " - 3 - 3

2 1 L 3 .e-6 3 3 3 6) 3 3 < 1 -3 9 3 + 1 3 9 I leolte-G 3 z 9 N 110.0498 +0.0025) g 3 2 0.2222 0:05 23 3 0.0174 20.2222 2 0.4048 .