U ~ $N(0,1)$, V ~ $N(0,\theta)$, W ~ $N(0,1)$ are independent random variables. $X = U+V$, $Y=V+W$, find the joint distribution of $(X,Y)$.

Question

U ~ $N(0,1)$, V ~ $N(0,\theta)$, W ~ $N(0,1)$ are independent random variables. $X = U+V$, $Y=V+W$, find the joint distribution of $(X,Y)$.

Here both $X \& Y$ are follows from $N(0,\theta+1)$, so they have the same distribution, then can I say the joint distribution is just one of PDF of them? Or the production of PDF of them?

Answer 1

You must prove that $X$and $Y$are independent for conclude that de joint is the multiplication of densities