Two random variables X and Y have bivariate normal distribution if the joint density is pdfX,Y (x, y) = 1 2xy 1

Two random variables X and Y have bivariate normal distribution if the joint density is pdfX,Y (x, y) = 1 2πσxσy √ 1 − ρ 2 exp [ − 1 2(1 − ρ 2) {(x − µx σx )2 − 2ρ x − µx σx y − µy σy + (y − µy σy )2}].

(a) Compute marginal probability density function of X.

(b) Show conditional distribution of Y given X = x is N(ν, τ 2 ) and find ν and τ .

(c) Show that X and Y are independent if and only if Cov(X, Y ) = 0.