Question

Let (X,Y) have joint pdf given by sey, 0 < x < y < 0, f(x, y) = { ( 0, 0.W., (a) Find the correlation coefficient px,y: (20 pts) (b) Are X and Y independent? Explain why. (10 pts)

Answer 1

fcay = ? -s é g Ocxcy coo Othererse on Marginal destribution of x . fo= ( for yg dag OD r 8 8 ote ole con of y. ng g [ 770 = otex fx x) = Marginal distribution of y. fy (8) = fcr,y) de & 4[x] - eg(y] =yer fy (Y) = gey olycos da é Now Ex = for fe as an = P x . ex dx - xex- & x 70 = Co-o) + 0 +0° = 1

Ey) = y. fyts = fag. yeč dy = yerdy = [-yh ey Jo + / 2 y et dy = 0 +2 [-yed-e7 ECY. = 2(0-0) +0+1] = 2 EXN) = x.g. ed dedy = $ſly egy + Jo syreddy? = 2 (0+ 360 y eddy ] - 3 pyte Yoly = ŞEM - 3 x 2 = 3. ELXY) =3

cov cx, 4) = E(X-y)-EXIE(Y) a concx y) = 3 - 1x2 = 3-2 = 1 Covcxy) = 1 Now ECx² = (xc². fx X dse = 1°R. et doe Ex² = 21 Now Elyy = So go fy cosay - Joggetdy 6 Ely) = from from the calculations of the calculation of E( XY). Varcx) = E3- EcuR 2-(1)2 = 24=1 Varcy = ECY} -(E(4)) ê 6-(2)² = 6-4 = 2

ox= oy= varcas Suarcy) = 5 J = 1 = Jr & corelation coefficant Pey = cov(x, y) = Tx de Svarx Jvary (6) since. Ex cas = et & fy cr= yey & fco,y) & fx cas. fyg) Therefore xey are not independent