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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?

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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".

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