Please see the problem in the attachment.

5. Wick's theorem (also Isserli's theorem)[10 points] Let X be a multivariate normal random vector with covariance matrix C. Use Wick's theorem (stated below) to evaluate E[X1XX4], E[X12X22] and E[X?]. Wick's theorem: Let X be a jointly Gaussian zeromean 2ndimensional vector with covariance matrix C . Then E(X1X2 . . -X2n) = E H E(Xin) 2H stands for summing over all distinct ways of partitioning the 2n random variables (possibly perfectly correlated, 1.6., X; = Xk for some l 7E 16) into n pairs of (Xi, X j), and each summand is the product of the n pairs. For example, if 2n = 4, we have E(X1X2X3X4) = 012034 + 013024 + 014023, and similarly, E(Xil) = E(X1X1X1X1) = 011011 + 011011 + C11011 2 3 [E(X12)]2