# How does jointly Gaussian random variables relate to Gaussian random variables?

Given two random variables $$X,Y$$, suppose that we know that they are jointly Gaussian.

Does the marginal distribution for $$X$$,$$Y$$ have to be Gaussians?

Conversely, given two random variables $$X,Y$$, each is Gaussian distributed, are they also jointly Gaussian?

Is there an intuitive reason for the answers?

Given two random variables 𝑋,𝑌, suppose that we know that they are jointly Gaussian.

Does the marginal distribution for 𝑋,𝑌 have to be Gaussians?

Yes - it is easy to see this simply by integrating out one of the variables from the pdf. Suppose $$\begin{bmatrix}x_1\\x_2\end{bmatrix} \sim N \Big(\begin{bmatrix}\mu_1\\ \mu_2\end{bmatrix}, \begin{bmatrix}\sigma_{11}, \sigma_{12}\\ \sigma_{21}, \sigma_{22} \end{bmatrix} \Big)$$

For convenience, we denote the mean vector as $$\mu$$ and the covariance matrix as $$\Sigma$$. Since we know that the pdf is given by:

$$(2\pi)^{-1} \text{det}(\Sigma)^{-\frac{1}{2})} \text{exp}( (x- \mu)^T \Sigma^{-1} (x- \mu)$$

If you integrate this with respect to $$x_2$$ (It's messy - Try doing this yourself by completing the square in the exponent), you will get the pdf of $$x_1$$, which is easily identifiable as the pdf of a Gaussian random variable.

Conversely, given two random variables 𝑋,𝑌, each is Gaussian distributed, are they also jointly Gaussian?

This statement is not true. Refer to https://en.wikipedia.org/wiki/Multivariate_normal_distribution for an example under the section "Two normally distributed random variables need not be jointly bivariate normal".