Assume $S$ is a $p \times p$ Wishart scatter matrix with the distribution $S \sim W_p (V, n)$. Let $\vec 1$ be an $p$ dimensional vector of ones, and let $A$ and $B$ be $p \times p$ symmetric positive-definite constant matrices. I have two questions:
What is the expected value and variance of the scalar $\vec 1^T (A + S)^{-1} V (B + S)^{-1} \vec 1$?
What is the expected value and variance of the scalar $\vec 1^T J (A + S)^{-1} V (B + S)^{-1} J \vec 1$, where $J = (t (A + S)^{-1} + (1 - t) (B + S)^{-1})^{-1}$ for some scalar $t$?
Even a partial answer would be very helpful.