Joint density of two vectors of multivariate normal random variables

Question

If $\bf{X}$ and $\bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $\bf{X}$ and $\bf{Y}$? Is it also multivariate normal?

Answer 1

As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X \sim \mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.

• Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z \sim \mathcal{N}(0,1)$.
• $X$ and $Z$ are uncorrelated.
• $X$ and $Z$ are not independent. (For example $P( |X| \le t, |Z| >t) =0$, but $P(|X| \le t) P(|Z|>t) \ne 0$ for $t>0$.)