Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

13but the most probable values:x, = p(x)dxx,., , - x, J*,In case of complicated density functions the determination of the limits of the equalprobability intervals themselves may be complicated and time consuming, however, one hasto bear in mind that this task is to be solved only once for a given PDF.Another restriction is that the method cannot be applied direct'functions defined over an infinite domain since the tail of the distributfif a table of finite dimensions is used. There are several cases whenbe overcome by tricky methods. A nice example was proposed b)(described in Reference 22) for the exponential distribution.Here, the PDF isp(x) = e "and samples can very easily be obtained by solution of the inverse distribution Equation(2.2),*x = P '(p) = -lnp (2 6)however, the execution of logarithm is very time consuming.Instead, let us write the realization of x in the formx = k • ln2 - z (2.7)where k > 0 and 0 < z =s ln2. Let us select a random number p and determine a value ksuch that2-x ^p=S 2" ( k f "It can be seen 30that choosing k in this way the cumulative distribution function of z inEquation (2.7) readsP(z) = e - 1 0 < z «: ]n2Since the random variable z is defined over a finite interval it can be selected by tablelookup and the z value so selected along with the integer k value above determines theexponentially distributed random variable x according to Equation (2.7).An actual realization of the above procedure is detailed in Reference 22.E. SELECTION FROM POWER FUNCTIONSLet p(x) = (n + l)x" O=Sx=Sl (2.8)* Here, the term (1-p) derived from Equation 2.2 is replaced by p since both are equidistributed on (0,1).