Multivariate Gaussian distribution

A multivariate Gaussian distribution has density proportional to $\exp [-(x-\mu {)}^T{\Sigma }^{-1}(x-\mu )/2]$. Its mean is $\mu$ and its covariance matrix is $\Sigma$.

For centered jointly Gaussian random variables, an odd product has zero expectation, and an even product's expectation is the sum over all pairings of the products of the pair covariances. In particular,For a proof, the moment-generating function is $\exp (t^T\Sigma t/2)$. Differentiating once in each variable at zero leaves precisely products of covariance entries indexed by complete pairings. No independence assumption between the coordinates is needed.

For a centered scalar normal random variable of variance $v$, the Isserlis theorem sums complete pairings of $2n$ factors. The first factor has $2n-1$ partners, then $2n-3$ choices remain for the first unpaired factor, giving $(2n-1)!!=(2n)!/(2^{n}n!)$. Odd moments vanish. Summing the moment-generating function gives $\exp (v{s}^2/2)$. This is a scalar formula, not a formula for the length of a vector.

Every linear combination $a^{T}X$ of the coordinates of a jointly normal random vector is a normal random variable. Its mean is $a^T\mu$ and its variance is $a^T\Sigma a$.