Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

416 Monte Carlo Particle Transport Methods; Neutron and Photon CalculationswithV 1(V 1 1)H 1(H 1Hj(Il 1- v) (7. 0 ) 2 'Dand therefore the unbiased estimate m reads2 '1,) 2 " (WS 1) 2 (1;,"/¾)instead of the maximum likelihood estimate corresponding to Equations (6.218) and (6.219).Comparison of the unbiased and maximum likelihood estimates leads to similar conclusions,as in the case of correlated rare sets, and will not be repeated here.D, ESTIMATION OF THE COMBINED VARIANCE OF RARE SETSFirst, we give an unbiased estimate of the variance of the estimated common mean interms of the rare sample statistics. Next, we show that if rarity is not realized and the whole -sample statistics are used for estimating the variance, then the estimate may be completelyunreliable. The considerations are first presented for correlated rare setsLet us denotec, = e T Q/e r Q reThen the unbiased estimate of the mean in Equations (6.234) and (6.235) readsm - PC nXHencerh — m = pc' r'(x - me/p) 4- m(p - p)/pwithP = v/nThe conditional variance of rh with v (the number of rare samples) given isD 2 lm|v] - p 2 D 2 [p>] + m 2 (p •- p) 2 /p 2whereD 2 [p>] = KE 1 1 Q 1*)]- 1Taking the expectation with respect to v, i.e., summing up with the binomial distributionof v, the unconditional variance becomesD 2 Im] = V ( 1 1 V(I - P)"-D 2 [m|v](6.248)- P - D 2 [p>] + m 2 (l - p)/(n,p)n