$\displaystyle \text{Let } Y \sim \mathcal{N}(\mu, 1) \text{ and } X|Y \sim \mathcal{N}(Y, \sigma^2) $.

$\displaystyle \mathcal{N} \text{ denotes normal distribution.} $

i) Show that (X, Y) pair has the same joint distribution as the (Y+Z, Y) pair, where Z is a independent standard normal random variable.

ii) Using i) show that X,Y pair is a two dimensional normal distribution.

iii) Calculate E(X), Var(X), Corr(X,Y).

iv) Find E(Y | X=x).

v) What is the conditional distribution of Y given X=x?

I would really appreciate if you could help me.