Bivariate Normal Distribution

Let (X, Y ) be Bivariate Normal with X N(0, ^2 1 ) and Y N(0, ^2 2 ) marginally and with Corr(X, Y ) = .

(a) Show that X, Y cX also follows a Bivariate Normal distribution for any constant c.

(b) Find a constant c such that Y cX is independent of X. ( Hint: Find c such that Cov(X, Y cX) = 0. Now argue why that also ensures independence.