<p><a href="https://stats.stackexchange.com/questions/245120/for-gaussian-distribution-matrix-x-find-efx-and-sigmax-s-t-f-sum">For Gaussian distribution matrix $X$, find $E(F(X))$ and $\sigma(X)$ s.t. $F=\sum f(x_i)P_i,$ given $E(f_i(x_i)),\sigma(f_i(x_i)) \forall i$</a></p>

<p>While jbowman answers my question, I am unsatisfied. I must know why his answer is true, and the clues he gave me in the edit have not been enough. Can I repost this question to the mathematics stack? Is there some etiquette to go through before I cross post?</p>
