Conditional PDF of Jointly Gaussian RVs [10+5+5 points] Suppose X and Y are jointly Gaussian random variables with correlation coefficient p. Let X~ N(ux,) and Y ~ N(uy, ). Let Z=E[X|Y], show that (a) Z = ux+%xp(Y UY). Hint: For bivariate Gaussian random vector (X,Y) with parameters given as above, the joint pdf is fx.y(x,y) = exp{2(1p)[(*Hx) + (V-HY) _ 2P(x-HX) (V-HY)]} 21 - (b) From part (a) conclude that E[XY] is a Gaussian random variable with mean x and variance xp. (c) We know that Z is the MMSE estimator of X from Y. Find the Mean Square Error of this estimator, i.e., E[(X Z)]. -