Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

on (0,1) and then take as the random numberythe value ofF−^1 (ξ). We now

illustrate this with a worked example.

Find an explicit formula that will generate a random numberydistributed on(−∞,∞)

according to the Cauchy distribution

f(y)dy=

(a

π

) dy

a^2 +y^2

,

given a random numberξuniformly distributed on(0,1).

The first task is to determine the indefinite integral:

F(y)=

∫y

−∞

(a

π

) dt

a^2 +t^2

=

1

π

tan−^1

y

a

+

1

2

.

Now, ifyis distributed as we wish thenF(y) is uniformly distributed on (0,1). This follows
from the fact that the derivative ofF(y)isf(y). We therefore setF(y)equaltoξand
obtain

ξ=

1

π

tan−^1

y

a

+

1

2

,

yielding

y=atan[π(ξ−^12 )].

This explicit formula shows how to change a random numberξdrawn from a population
uniformly distributed on (0,1) into a random numberydistributed according to the
Cauchy distribution.

Look-up tables operate as described below for cumulative distributionsF(y)

that are non-invertible, i.e.F−^1 (y) cannot be expressed in closed form. They

are especially useful if many random numbers are needed but great sampling

accuracy is not essential. The method for anN-entry table can be summarised as

follows. DefinewmbyF(wm)=m/Nform=1, 2 ,...,N, and store a table of

y(m)=^12 (wm+wm− 1 ).

As each random numberyis needed, calculatekas the integral part ofNξand

takeyas given byy(k).

Normally, such a look-up table would have to be used for generating random

numbers with a Gaussian distribution, as the cumulative integral of a Gaussian is

non-invertible. It would be, in essence, table 30.3, with the roles of argument and

value interchanged. In this particular case, an alternative, based on the central

limit theorem, can be considered.

Withξigenerated in the usual way, i.e. uniformly distributed on the interval

0 ≤ξ<1, the random variable

y=

∑n

i=1

ξi−^12 n (27.55)

is normally distributed with mean 0 and variancen/12 whennis large. This

approach does produce a continuous spectrum of possible values fory, but needs