Suppose $U $ and $V$ are independent and each is distributed as $ N(0,1$). Define $ X$ and $Y$ by $Y=X-1-U$,$ X=2Y-3-V$ . What is $E(Y|X)$ and Var$(Y|X)$ ?
Again another questions which I'm unsure about.
I know the formula of $E(Y|X)$ and $Var (Y|X)$ if $f(x,y)$ is a bivariate normal distribution but I have a feeling I can't assume that here.
So would I have to do a transform of variables to get the answer? When I did so, I have trouble doing the integration of $f(x,y)$ to find the marginal density $f(x)$.
$(U,V)$ is bivariate normal, and $(X,Y)$ is obtained from that by an affine transformation, so $(X,Y)$ is also bivariate normal.