#conditioning

Question 1: Select all that apply

Gaussian process inference relies on closure properties of Gaussian random variables. Which statements are true? Select all that apply.

If $X$ and $Y$ are independent Gaussians, $X + Y$ is Gaussian1
If $X$ and $Y$ are independent standard Gaussians, the product $XY$ is Gaussian2
If $(a, b)$ is jointly Gaussian, then $a$ given $b = \beta$ is Gaussian for every $\beta$ 3
If $(a, b)$ is jointly Gaussian, then the marginal distribution of $a$ is Gaussian4
If $a$ and $b$ are each Gaussian, then the pair $(a, b)$ is jointly Gaussian5