TABLE OF CONTENTS This section introduces the general scheme of the MC method and outlines its application to the solution of integrals and integral equations.
To calculate some unknown value m by the MC method one has to find a random variable ? whose expected value equals E{ ?} = m. The variance of ? is designated Var { ?} = ? 2 with ? being the standard deviation.
Consider N independent random variables ? 1, ? 2, , ? N with distributions identical to that of ?. Consequently, their expected values and their variance coincide
Expected value and variance of the sum of all these random variables are given by
Using the properties E{ c ?} = cE{ ?} and Var{ c ?} = c 2Var{ ?}, one obtains from (2) and (3)
Therefore, the random variable
has the same expected value as ? and an N times reduced variance. A MC simulation of the unknown m consists of drawing one random number ?. Indeed, this is equivalent to drawing N values of the random variable ?, and evaluating the sample mean (6).
The MC method gives an estimate of both the result and the error. According to the central limit theorem the sum ? N = ? 1 + ? 2 +
