Question

Let (X, Y) have joint pdf given by 0 y 00, ey f(x, y) 0, o.w., (a) Find the correlation coefficient px,y. (20 pts) (b) Are X and Y independent? Explain why. (10 pts)

(a) Find the correlation coefficient ρX,Y .

(b) Are X and Y independent? Explain why.

Answer 1

Šll le-y; ocalycos flory)=2 1 0 ; otherwise – =3 Too on y dy (1) Marginal pid-f ofx, f(x)= l floriydy = y = x y=x 318)= (-eyle=e-x ie free e-x; OLXco Marginal pidif- of y, y fly)= [ f(x,y)dx= 2-0 fry)= yer olycosas ru ylidae Coy dre=en song in formue ferdsefine XED 24-0 Hence, Elx) = fx fra de feºda Elgeta famigas des fite des 2 = 2 2 - 2

y eddy=12=2 [sie 5 mes* due = 16 1] * also cry)= P4-819) dy fy yetene for everyou-2 Elva)= (y2 fin ddy a fyzye'dy jöven dy=[3 = 6 Mence vas(x) = E(X?) – [E{x)) = 2-032= 1 Vaz (Y)= Ely?) – [Ely) = 6-(2)2 = 2 Cor (X,Y)= E(XY) – EXX). ECY) EXXY) = S(xy.frxiy) andy ay.ér.dy dre 2150 YX YEX d=0 - ful fretende jeffe conf- fe, 1954de Мех ЯРХ

Po =D trave [a foeteye fora, a - felves* pere) dhe suce you to get to (14) -o) Txe x=0 - = (2+ L = 2+1=3 ftet bos frete = 0+1 = 293 1 x ² x dat lae dx x=0 2100 ice ExY)= 3 Cor(x, y) = E(XY) – E(X). Fy)= 3-(1)(2-) = || Cov(x, y) = Vas (x) varly) = Tz Pex,y) V(17.(2) Anshee (b) f(x). fly)= el yey +/, y) : e-> sunce fro,y) + f(e). ffy) Theis y & y are not widebendout a = Aushan