Finding the variance of a multivariate Gaussian

Question

I have a multivariate Gaussian random vector $x \sim (p, \Sigma)$. Now, is there a way to find the variance of the random variable $x^T\Sigma^{-1}p$?

Answer 1

If $x$ is an $n$-dimensional MVN and $A$ is a constant $m\times n$ matrix then the covariance matrix of $Ax$ is given by $$ E(Ax(Ax)^T) -E(Ax)E((Ax)^T) = E(Axx^TA^T)-E(Ax)E(x^TA^T) \\ = A(E(xx^T)-E(x)E(x^T))A^T = A\Sigma A^T.$$

So since $x^T\Sigma^{-1}p = p^T\Sigma^{-1}x,$ we can apply the above formula to get the variance you want.