Suppose X and Y are jointly Gaussian random variables with correlation coef- ficient . Let X N X, 2 X and Y N Y , 2 Y . Let Z = E[X | Y ], show that (a) Z = X X Y (Y Y ). Hint: For bivariate Gaussian random vector (X, Y ) with parameters given as above, the joint pdf is fX,Y (x, y) = exp n 1 2(12) h (xX) 2 2 X (yY ) 2 2 Y 2(xX)(yY ) XY io 2XY p 1 2 (b) From part (a) conclude that E[X | Y ] is a Gaussian random variable with mean X and variance 2 X