Question

Let (X,Y) have joint pdf given by f(x, y) = { Sey, 0 < x <y<, | 0, 0.W., (a) Find the correlation coefficient px,y (b) Are X and Y independent? Explain why.

Answer 1

Given, , ens t(m,y} 0<x< y<co Toi on $l9 = {**my day = {"c%dy fyloge se delle Leylasili - = yey, oxyla A X E[X] - J* * 54 m) da E[x]- = 620 * - 65). = 0 - 10 - 1) = 1 EDI] = story fyly) dan = (-y2e-- 280 -y - 26-ylo - O - FO-0 - 2) = 2.

so, Elxo) - 7">"; ea da. .aren - onen ze and e[y] Somo ya. yeay dy (-y) e-o - sykevy - bye-y-be-yllo = = 6: va xo(x)= E(**) - (E (*)) 2. 2-1=1 vonly) - 6Y) - (EY)) = 6 - 2 - 2. @ Cov (234) E(XY) - E(X) ELV) Now, E(XY) = end du dy < e i = - 60 de day. Joan ys eny de 1 x6 = 3. Cote (x, ) = 3 1x2 = 1 P (x,y), Coco (,y) Tvarow) vanly) rizs © fyin) fyly) - e-*. yeyeye f(y). i. xf Y ane not independent.
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