Gaussian if their joint PDF is given by fx.x CxP for(sx) - 20ar(1-Phy) (-21-P) [(-)-(-~^) (-)-()]). + This PDF is specified by 5 parameters which correspond to the marginal means, variances and correlation, i.e.. (X,Y) ~ N(HX HY, 00 Px Some interesting facts about bivariate Gaussians are listed below: Marginals are Gaussian: X~ N(ux,) and X~ N(x,0). Bivariate Gaussian RVS are uncorrelated iff they are independent: If pxy=0 then the joint density can be written as -(x-x)/20 -e-(x-)/2017, fxx (x,y) = fx(x) fy (y) = V2 fxy(x,y) fx (x) 1 V 2, thus the RVS are independent. A linear transformation of bivariate Gaussian is also bivariate Gaussian, but with new parameters. Bivariate Gaussian and conditioning: For bivariate Gaussian RVs (X,Y) the conditional distributions fxy and frix are also Gaussian, and are given by, fy|x (v|x)= where E[Y|X] =Myx (X) = y +Pxy X (X-Mx) and Oyx = o(1-P). Suppose that X~ N(0, 1) and Y~N(1,2) are two independent Gaussian random variables. Define random variables U and V as U=X+Y, V=X+Y for some a 1. What is the joint distribution of U, V in terms of a? 2. Find a (if it exists) such that U and V be independent. 3. Find the conditional distribution fulv (u5). exp 2Toy MY _(x X and Y are bivariate ( HY\x(x))2 - Hyx (x)))) ~ N(M\x, 07 (x), 20|x Suppose that X ~ N(0,1) and Y~ N(1,2) are two independent Gaussian random variables. Define random variables U and V as U = X + Y, V = x + ay for some a. 1. What is the joint distribution of U, V in terms of a? 2. Find a (if it exists) such that U and V be independent 3. Find the conditional distribution fu | v(u | 5).