Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

435ThenT.m = EJ.Iy.y.Rjeyiy,,,and since y is normally distributed, 1we have = S 13S 1,,, + S 11S 11n+ s, n!s,S 1J being the (ij)-th element of S. Accordingly(T 1111) = 2 S 13(R 11+ R 11)S,,,, + S 11n2 S 11R,.iijiwhich calls forth Equation (C.4).APPENDIX 61):EMPIRICAL THIRD MOMENTSLet us consider three correlated random variables u,v, and w. We can assume withoutloss of generality that the variables all have zero expectations, i.e.,(u) = (v) = (w) = 0Let U 1, Vj, and W 1( i = 1, 2 n) be independent realizations of the respective variablesand let the empirical means of the realizations be1 " 1 n I -\ii -- E ",- v = - VV ;, w =• - 2 w, (D !n i=i n 1 =i ni =.We wish to construct an unbiased estimate of the third momentT = (uvw)(D.2)Theorem — The estimateT = E (u, - u)(v, - v)(w, - w)/[n(n - l)(n - 2)] (D.3); •= iis unbiased with respect to T in Equation (D.2).Proof.It is to be seen that(T) =TTaking the expectation of Equation (D.3), we haveit(T) = [n(n — l)(n — 2)] " l S ( u . v i w .. ~ U 1V 1W — U 1VW 1- Uv 1W;i = i+ u.vw + uv,w + uvw, — uvw) (L).4)