Question

Given f(x,y) = 2 ; 0 <X<y< 1 a. Prove that f(x,y) is a joint pdf b. Find the correlation coefficient of X and Y

Answer 1

In this for obtaining the correlation we first find marginal distribution function and then using marginal we can obtain the covariance and hence correlation can be obtained. Here the limit of x and y depend on one another .

frnigd is a pdf Xy We have flu,y) = 2 of us yaa. 9. If 5 sf (my) obedy Boty - Y=2 > 1 [may (1) de 25 adu dy = fa 2 dn sady = 12 2du (1-x) = 6 (aus de = 2(x - x)] = 2 [1-4] sex(1-1) = Ox = 1 so & may) is a pdf. b) Corr(x, y) = Coulxit) Tvar (x) war (7) first we obtain marginal density of x and Y. Exlu) = S tady = 9 (2-2) ; OPHT 1 tyly) = jo z du ; organ Edy i ory! I {(x) = 2 a. fx (u) du = 1 . 2 (2-2) de | 3 x - y) - = a ( 1) okres

- 2x(4-5) = 2* (34) = {(t) = 1 tyly) ay = 1 & 2 yo y dy u(x) = E(X2) - É() 12 {(x²) = 52 m². fx (m) dhe = $.261-10) du = $50 (m?- xSchen =28.2 (n?_?)du = ax Comoros = 2x(5-1) = 2X E(79) =shga fylgdy = Say?. Gydy = 2 x, ypdy = 2 x ( = 3 e VC) = E(x2 – TECX)? = 4-6312 = 0.085.

N(Y) = = {(7) - (E(Y) 22 -3)2 = 0.5-0.444 - 0.055. Also, {(xx) = go dt (xxy) fxx (M14) don day = 8 2 1h sry 12, dn dy = 5² 27j ydy ) ah =st on est de conx (1–12) din = 2 x (1-2) du = 32 CM-23) du = [ 2 - 2 12 = ( 2 2 2 2 / 4 = À 20:25 Cou(x, y) = {(XY}-{(x). E(Y) -0.25-0.02222. = 0.0277 Corr(x,y) = Cou(xi) ✓ vai (x). Var(Y) - 0.5036. – 0.0 277 0.0277 So 055x0.085