# Thread: Joint distribution, conditional distribution

1. ## Joint distribution, conditional distribution

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