Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

427Nowi'6.77K)kj„,and therefore(V 2 ) = -kjfBy analogy to the derivation of Equation (6.278J. we have from Equation (6.276) that = - (x 2 ) + - - m 2 = - D 2 Ix] + irr (6.279)n n nand therefore(V 2 ) = [(x 2 ) - m 2 ] = ~ D 2 Ix]knNirrespective of the number n of histories in a batch. We have thus demonstrated that V 2inEquation (6.277) is an unbiased estimate of 1/N times the theoretical variance of x. Therefore,denotingS 1= (mj - ha) 2 , J= 1,2, k, (2«k«N)k — 1Sj represents a sample in the estimation of D 2 [x]. The corresponding sample mean is.1ks = r S Sj (6.280)k j_iand(s> = D 2 [x]It is, however, not necessarily true any more that for a given number of histories, thereliability of the variance estimate in Equation (6.280) is independent of the number ofbatches (or, equivalently, of the number of histories in a batch). Therefore, if the varianceof x is the quantity to be estimated, then the scheme resulting in the lowest variance of theestimated variance is to be applied. The most reliable estimate is defined in the followingtheorem.Theorem 6.14 — The variance of the sample variance is minimum if every batchcontains a single history, i.e., if n = 1 and k = N.Proof. The variance of the estimate s in Equation (6.280) is