Consider continuous random vector (X,Y) where X N(0,1) (standard normal) and Y |X = x N (x, 1) (the conditional distribution of Y given X = x is normal with mean x and variance 1).

1. (a)Find the joint pdf of (X, Y ).
2. (b)Find the marginal distribution of Y .
3. (c)Considerrandomvector(V,W)whereW N(0,2)(normalwithmean0andvariance 2). How should the conditional distribution V |W = w be chosen so that (V, W ) has the same distribution as (X, Y )?
4. Hint: Note that "(V,W) has the same distribution as (X,Y)" implies that for any v, w R.
5. fV,W (v, w) = fX,Y (v, w), (1) where fX,Y (, ) is given in part (a). In addition,

fV,W(v,w)=fW(w)f(v|w)= 1 2 2

Now you solve (1) and (2) to find f(v|w).