Joint distribution of two marginal normal random variables

Question

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) \end{align*} and let $Y = X_1 - 3X_2$, then what is the joint distribution of $X_1$ and $Y$?

My working: The marginal distribution of $X_1$ is: \begin{align*} X_1 \sim N(6, 12) \end{align*}

The marginal distribution of Y is: $$Y \sim N\left(-3, 12 \right)$$ since \begin{gather*} E\left(Y\right) = 6 - (3)(3) = -3 \\ V\left( Y\right) = 12+ (-3)(2)(-3) + 2(3)(-3) = 12 \end{gather*}

So we have $X_1 \sim N\left(6, 12\right)$ and $Y \sim N\left(-3, 12\right)$, first we need to work out the covariance between $X_1$ and $Y$: \begin{align*} cov\left(X_1,Y\right) & = E(X_1Y) - E(X_1)E(Y) \\ & = E\left(X_1\left(X_1-3X_2 \right) \right) - 6(-3) \\ & = E(X_1^2) - 3E(X_1X_2) + 18 \\ & = V(X_1) + E(X_1)^2 -3\left[cov(X_1, X_2) + E(X_1)E(X_2) \right] + 18 \\ & = 12+36-3\left[3+(6)(3) \right] + 18\\ & = 3 \end{align*} Hence, the joint distribution between $X_1$ and $Y$ is given by: \begin{align*} \begin{bmatrix} X_1 \\ Y \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ -3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 12 \end{bmatrix} \right) \end{align*}

Query:

I feel my answer is not right because I assumed that if we have two marginal normal random variables, say, $X_1 \sim N(\mu_1, \Sigma_{11})$ and $X_2 \sim N(\mu_2, \Sigma_{22})$, then $X_1$ and $X_2$ is jointly normal. That is,

\begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \Sigma_{11} & cov(X_1, X_2) \\ cov(X_1, X_2) & \Sigma_{22} \end{bmatrix} \right) \end{align*} where $cov(X_1, X_2)$ denotes the covariance between $X_1$ and $X_2$.

However, I know in general just because $X_1$ and $X_2$ are both marginally normal, this does not mean they are jointly normal. How else can I find the joint distribution between $X_1$ and $Y$?

Thanks

Answer 1

Recall the following fundamental fact about normal distributions:

If a random vector $X$ is normal $(M,\Sigma)$ then, for every matrix $A$, $AX$ is normal $(AM,A\Sigma A^T)$.

In your case, $A=\begin{pmatrix}1 & 0\\ 1 & -3\end{pmatrix}$ and $X=\begin{pmatrix}X_1\\ X_2\end{pmatrix}$ yield $AX=\begin{pmatrix}X_1\\ Y\end{pmatrix}$ hence you simply must compute the mean vector $AM$ and the covariance matrix $A\Sigma A^T$.

Note that the hypothesis in this exercise is not that $X_1$ and $X_2$ are normal but that the random vector $X=\begin{pmatrix}X_1\\ X_2\end{pmatrix}$ is normal. In the former situation, you are right to stress that chaos could ensue, but not in the latter.