Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

420 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsandy = ; ~ 2 y,nIn the derivations below, we shall assume that the variables are indeed simple arithmeticaverages, although generalization to more general types of sample means (e.g., such asthose introduced in Section A) is straightforward.Letf = y/x (6.257)f is a most reasonable but usually biased estimate of r in Equations (6.256) and (6.257).Now, let f(x,y) be the probability density function of the random variables x and y. If thesevariables are sample means from a sufficiently large number of realizations (i.e., if n » 1),then (except for unusually extreme distributions) according to the central limit theorem, thedensity function f(x,y) is dominantly concentrated on a small domain:A = jxe [m, — a,,ITi 1+a,]; y € [m 2-a 2,rn 2+ a :J}; 0 < a, < m ((6.258)f(x,y) = 0 if (x,y) I A (6.259)Note that this assumption is essential because if arbitrary values of x (including zero) wereallowed, then the expectation of y/x would diverge. Although this divergence may, inprinciple, occur, for sufficiently large samples, the probability of its occurrence is negligible.A simplified analytical example will demonstrate the nature of this assumption at the endof this section.The expectation of the estimated ratio in Equation (6.257) is(r) = J jixj