Question

2. (30 Points) X and Y ~ N (0,4) are two jointly Gaussian random variables, and E(XY) = 3 a. (10 Points) Find their joint PDF, f (x,y). b. (10 Points) Find the mean and variance of Z = X +Y. c. (10 Points) Find the mean and variance of Z = X + Y + 2.

Answer 1

Xu N10A) Yu No,4) ECXY)=3 cov (X,Y)= E(X4) - Elx) Elu oxo = 3 - cov (X,Y) = 3 ca) correlation coeff. (o)= concx, 4) NVIRU14) fiw ✓ 4x4 Joint pat ot x and ElxI=14 4 is ich v (x1= n V (Y) = 2 VH=0 fx, y (2,4)= I TR 6,62 N1-02 exp- . 20 (2-1) (9- GO + (9-4) 2 } but m=0 = 4 U2=0 2 =4 P = 3 Exy (x,y) = t - expll (1-2 0-0,² - 2x3 27 IA. 2.2 *2.2 V1212 4 X2x2 inte exb4 { **** Jay + 2yº? = exp- / (2x²-3 ay + 2y 2).

© Z = x+y meam of Z E12)= E(X) + EN EIZ) = 0 COVKM)=3. Variance of 2 V(2)= v(x)+ VH+ 220V (X4) = 4+4 + 2x 3 - V (Z) = 14 zu No, 14) to Z = x +4 + Ź meonot z EIZ) = E(X) + E142+2 otot2 ELZ) = 2 V (Z) - Vexstva) + 2 Cov(x4) = 4 + 4 + 2x3 V(z) = 14 zu w 102, 14)