Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSon (0, 1) and then take as the random number y the value of F −1 (ξ). We nowillustrate this with a worked example.◮Find an explicit formula that will generate a random number y distributed on (−∞, ∞)according to the Cauchy distribution( a)dyf(y) dy =π a 2 + y , 2given a random number ξ uniformly distributed on (0, 1).The first task is to determine the indefinite integral:∫ y ( a)dtF(y) =−∞ π a 2 + t = 1 y 2 π tan−1 a + 1 2 .Now, if y is distributed as we wish then F(y) is uniformly distributed on (0, 1). This followsfrom the fact that the derivative of F(y) isf(y). We therefore set F(y) equaltoξ andobtainξ = 1 y π tan−1 a + 1 2 ,yieldingy = a tan[π(ξ − 1 )]. 2This explicit formula shows how to change a random number ξ drawn from a populationuniformly distributed on (0, 1) into a random number y distributed according to theCauchy distribution. ◭Look-up tables operate as described below for cumulative distributions F(y)that are non-invertible, i.e. F −1 (y) cannot be expressed in closed form. Theyare especially useful if many random numbers are needed but great samplingaccuracy is not essential. The method for an N-entry table can be summarised asfollows. Define w m by F(w m )=m/N for m =1, 2,...,N, and store a table ofy(m) = 1 2 (w m + w m−1 ).As each random number y is needed, calculate k as the integral part of Nξ andtake y as given by y(k).Normally, such a look-up table would have to be used for generating randomnumbers with a Gaussian distribution, as the cumulative integral of a Gaussian isnon-invertible. It would be, in essence, table 30.3, with the roles of argument andvalue interchanged. In this particular case, an alternative, based on the centrallimit theorem, can be considered.With ξ i generated in the usual way, i.e. uniformly distributed on the interval0 ≤ ξ