Question

4. Suppose X and Y has joint density f(x, y) = 2 for () < x <y<1. (a) Find P(Y - X > 2). (b) Find the marginal densities of X and Y. (c) Find E(X), E(Y), Var(X), Var(Y), Cov(X,Y)

Answer 1

ADG Let xf y Froery)=2 has joint deosity for o coecycl @ - Marginal densities of xfy fx loe) = f free.y) dy - Ex (se) = f 2 dy - 2 (1) de - Exfoe) = 2(1-2); ocael ft 19 = 2 de tycy) - 2.Czen! - fy (4) & (L) = 2 joc 0 EcXI= afcae) are se ex) = 1'2ci-zes dae E(X)= lo'zda-lo zadee E(X) = 2(x2 - 2 (22)

€ (x) = 2 [(-o)]-[C1-09) E(X)= 2 EX)= ] 2-1=1 E (X) = sode²2 (1-2e) dae Ecxe)21. ?dze - Go' sedae €(22)= 2 ( Lagyo - (] Ecx)= ?( (1-0) – +(1-0)] Ecx!). 2(t-) ecx²) = e( Ex) = 16 var() ex)- (EC) vaxcx) = ely soy (2) dy Ech) - 2 (2) - € (7²) - soy2 zdy exy = 26)

varly)= E(4) - (ECU)) ². varly Var (Y) = 2 varra) - varry exy) - 11 ay 2d sedy ext) eso y te dy ext) 25'(92-0)dy e(x4) = sol (92) dy E(XY) = siyady € (44) = (4.4)! E(XY)= Covexy ) = Eixy) F(X) Ely) Courty) = 2 - 1 - 2