I'm having trouble with this question from a past qualifying exam:
Question
Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. Define two random variables:
$X = aZ + bW$ and $Y = cX + dV = caZ + cbW + dV$
Find the conditional density function of $X | Y = y$.
Attempt At Solution
We know by property of normal RV's that:
$X \sim N(a\mu, a^{2}\sigma^{2}+b^{2})$ $\;$ and $Y \sim N(ac\mu, a^{2}c^{2}\sigma^{2} + c^{2}b^{2}+d^{2})$
Are $\begin{pmatrix} X \\ Y \\ \end{pmatrix}$ then jointly normal? If so, then I think I can proceed without problem. However, I'm not sure if marginal normals imply jointly (bivariate) normal.
Can someone shed light on this for me? Thanks in advance.