Joint distribution of subset of jointly Gaussian random variables I have a random variable $X$, which is a $d$-dimensional vector that varies with a multivariate Guassian $X \sim \mathcal{N}(\mu, \Sigma)$, and I have a set $S \subseteq \{X_1, X_2, \dots, X_d\}$, which is a subset of the components of $X$. what is the joint pdf of that subset: $f_{S}(x)$? 
For example, if I had $d=3$ components of $X$, what would the joint pdf be of just $X_1$ and $X_3$, $f_{X_1, X_3}(x)$ be? My guess is that the pdf is also a Gaussian with the corresponding entries of the mean vector and covariance matrix, but I don't have a real proof of this. There is a proof for the bivariate case on the first page of this.
 A: HINT
Think about the characteristic function. You have $$ f_{(X_1,X_3)}(t_1,t_3) = E(e^{i(t_1X_1+t_3X_3)})$$ which is just the characteristic function of $(X_1,X_2,X_3)$ with the variable $t_2$ set to zero. Then use the form of the characteristic function for a multivariate Gaussian.
(Then this can be generalized to any subset.)
A: As in my previous answer, it will be convenient to operate under the following definition of the Gaussian distribution.

$X$ is defined to be multivariate Gaussian if and only if $c_1 X_1 + \cdots + c_d X_d$ is Gaussian for any scalars $c_1,\ldots,c_d$.


Using the above definition of the Gaussian distribution, it is easy to check that the vector $(X_i)_{i \in S}$ is Gaussian. Indeed, any linear combination of the $X_i$ where $i \in S$ is also a linear combination of the full collection $X_1,\ldots,X_d$, which by definition is Gaussian.
Therefore, it just remains to find the mean vector and covariance matrix for $(X_i)_{i \in S}$, which can be read off $\mu$ and $\Sigma$; specifically, it is the sub-vector of $\mu$ and sub-matrix of $\Sigma$ indexed by $S$, which is what I think you guessed. Knowing that $(X_i)_{i \in S}$ is Gaussian and having computed its mean and covariance matrix, you can write down the pdf.

Caveat: Note that the above solution (as well as the characteristic function approach) are embarrassingly simple. However, if you are doing homework and are not allowed to use either of those definitions/characterizations, then you will need to prove the equivalence of the pdf definition of the Gaussian distribution with one of the other definitions (linear combinations are Gaussian or characteristic function), which may be a little annoying. This caveat goes for my previous answer as well.
