Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

41?(Note that vD 2 [|i|v] is independent of v; that is why it can appear in the formula afteraveraging over v.) Now. the results in Section B concerning notirarc .-.els apply Ui thevariance of p.(since the rare samples themselves are assumed to be Gaussian), i.e., denotingV 2 = lv(e T Q re)l-'an unbiased estimate of the conditional variance D 2 J |i|id follows from Equation (6.227;^ V 2 Ii + 2(k - l)/(v - D]Since u, = mZp, the second term on the RHS of Equation (6.248) can be rewritten asnr(l - p)Z(np) = LI-'U - p)p/n (6.249)We have seen in the previous section that p. can be estimated according to Equations (6.2"¼ t.i.e.,u, 2 « p 2 = A 2 ZfD 2 (6,250¾To complete the derivation, we need an unbiased estimate of p(ishows that— p). Simple algebra(P(I - P))Z(n - 1) = P(I - P)Zn (6.23 J)i.e., np (1 - p)/(n - 1) is an estimate of p (1 - p). Accordingly, the second term on theRHS of Equation (6.248) is estimated asm 2 (l - p)Ztp(n - 1)1and we have the following theorem.Theorem 6.11 — An estimate unbiased up to the order (k/np) Jof the rare-sampkvarianceof the common sample mean rh in the case of correlated rare events isS r= m 2 (l - p)/[p(n - 1)] + p 2 V 2 [i + 2(k - !)/(*> - !)] (6.257)Naturally, if p = 1 (i.e., if v = n), Equation (6.252) goes over to Equatihowever, p is considerably less than unity, Equation (6.252) defines an estimate i, hi -from that in Equation (6.227). This will be shown below. The (corrected] Gainlikelihood estimate of the variance of m* (i.e., of the whole-sample combii" 1i t •-.iEquation (6.240) would readS* = In(CrQe)I- 1 Il + 2(k - l)Z(n - 1)]Simple but lengthy algebra shows 49thatIe 1 QeI1== p[e T Q,.e]1+ m=(l - p)/(pnwhere r is the rarity factor defined in Equation (6.242).