Joint Gaussian distribution implies Gaussian + independence?

Question

Assume that $X = (X_1, ... , X_n)$ are jointly Gaussian distributed.

Can I then say that each of $X_1, ... , X_n$ is Gaussian distributed?

Can I deduce that they are pairwise independent?

Answer 1

As has already been noted, the marginal distribution of each $X_i$ will be Gaussian. But the $X_i$ are independent iff they're uncorrelated. This is a very convenient result, but it does require the joint distribution to be multivariate Gaussian; it doesn't follow from the marginal distributions being Gaussian.

Answer 2

If random vector $\vec X=(X_1,\dots,X_n)^T$ has normal distribution then $A\vec X$ has normal distribution (here $A$ denotes $m\times n$-matrix with entries in $\mathbb R$).

Applying that with $A=A^{1\times n}=(0,\dots,0,1,0,\dots,0)$ we find that the $X_i$ have normal distribution.

The $X_i$ are not necessarily independent.