TABLE OF CONTENTS This chapter introduces the concept of Monte Carlo integration and reviews some basic concepts in probability theory. We also present techniques to create better distributions of samples. More details on Monte Carlo methods can be found in Kalos and Whitlock [86], Hammersley and Handscomb [62], and Spanier and Gelbard [183]. References on quasi Monte Carlo methods include Niederreiter [132].
The term Monte Carlo was coined in the 1940s, at the advent of electronic computing, to describe mathematical techniques that use statistical sampling to simulate phenomena or evaluate values of functions. These techniques were originally devised to simulate neutron transport by scientists such as Stanislaw Ulam, John von Neumann, and Nicholas Metropolis, among others, who were working on the development of nuclear weapons. However, early examples of computations that can be defined as Monte Carlo exist, though without the use of computers to draw samples. One of the earliest documented examples of a Monte Carlo computation was done by Comte de Buffon in 1677. He conducted an experiment in which a needle of length L was thrown at random on a horizontal plane with lines drawn at a distance d apart ( d > L). He repeated the experiment many times to estimate the probability P that the needle would intersect one of these lines. He also analytically evaluated P as
Laplace later suggested that this technique of repeated experimentation could be used to compute an estimated value of ?. Kalos and Whitlock...
