Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

63In the following Sections, an elementary treatment of the statistical uncertainties isgiven. Derivations of the formulae given as well as more thorough discussion of their validitycan be found in many textbooks on probability theory and statistics." 16 - 29 - 40A. THE CENTRAL LIMIT THEOREMLet us consider n independent random observations, p.,, jx 2, . . . , ix nof a randomvariable. Assuming that this random variable (cp) is a function of t, with a PDF p(t), theexpected value of cp is defined byM(cp) = f cp(t)p(t) dt(3.35¾The real meaning of t in our case is quite general, it symbolizes a variable by which all thepossible random paths can be parametrized.The variance is defined as:D 2 (cp) = M(Cp 2 ) - [M(Cp)] 2If one estimates the expected value by the average of the n samples, i.e., byM(cp) = M-,1 "X M-,n , ,then according to the law of large numbers the average |i approaches the expected valueM(cp) with a probability that approaches 1 as the sample size increases (n —» °°).More precise information on the convergency of the estimation is given by the centrallimit theorem.Given the n observations described abovelim Pa^ Vn ^ b] = -j= Pe- 2 ' 2 dt (3.36)D(cp) J \/2tt J.where P{x} denotes the probability that x is true. Equation (3.36) means that the average ofn independent observations of a random variable (with finite mean and variance) approachesa normal distribution.Substituting a = — 1 and b = 1 into Equation (3.36); the probability thatIfI n- M(cp)| > D(cp)/\/nis about 32%. The probability that, for example, the difference between the average andthe expected value exceeds 3D/\/n is only 0.27%.In practice Equation (3.36) is not directly evaluable since the variance D 2 (cp) is notknown in advance. A method to estimate it is given in the next Section.IL THE ACTUAL COMPUTATIONSAt the actual computations histories of n particles are followed resulting in scoresM-!, u-2> • • • ,M-n- The definition of the average is: