Proving $(X,Y)$ is a normal vector when $X\sim N(1,1)$ and $Y\mid X\sim N(3X,4)$

Question

Suppose I have a random vector $(X,Y)$ with $X\sim\mathcal{N}(1,1)$ and $Y|X = x \sim\mathcal{N}(3x,4)$.

I need to prove that $(X,Y)$ is a normal vector as well.

To do that I want to explicitly write the vector of expected values and the $2x2$ matrix of variance and covariance.

I know that the first entry of the vector of expected values is 1 and the entry $C_{1,1}$ of the matrix is 1 as well. However I am struggling to see how I can derive the density function of Y from the conditional one.

Any suggestions?

Answer 1

You could say $Y=3X+Z$ where $Z\sim N(0,4)$ independent of $X$.

It is then an easy step to say that $Y\sim N(3\times 1+0,3^2\times 1+4)$ and

• $E[X]=1$
• $E[Y]=3$
• $\text{Var}(X)=1$
• $\text{Var}(Y)=13$
• $\text{Cov}(X,Y)=\text{Cov}(X,3X)=3\text{Var}(X)=3$

Answer 2

You can proceed with moment generating functions.

Joint MGF of $(X,Y)$ is

\begin{align} M(s,t)&=E\left[e^{sX+tY}\right] \\&=E\left[E(e^{sX+tY}\mid X)\right] \\&=E\left[e^{sX}E(e^{tY}\mid X)\right] \end{align}

From the given information, you should be able to show that this MGF corresponds to the MGF of a bivariate normal distribution. That would complete your proof using the uniqueness property of MGF.