(Solved): Suppose X N(0, 1). Suppose [Y | X = x] N(x, 1 2 ), where (1, 1). (1) Find ...

Suppose X ∼ N(0, 1). Suppose [Y | X = x] ∼ N(ρx, 1 − ρ 2 ), where ρ ∈ (−1, 1). (1) Find f(x), f(y|x), and f(x, y). Calculate E(Y |X = x) and Var(Y |X = x). (3) We can also write Y = ρX + ϵ, where ϵ ∼ N(0, 1 − ρ 2 ), and ϵ is independent of X. Find E(Y ), Var(Y ), and Cov(X, Y ).