To understand the mathematical basis of Monte Carlo calculations, to develop means of increasing the efficiency of such calculations, and to estimate the statistical uncertainty in the results obtained, it is necessary to have some understanding of statistics and probability theory. A brief description of the terminology that will be used in the remainder of this book is also necessary. The intent of this chapter is to introduce the mathematical concepts that underlie the Monte Carlo method and provide a basis for further development of selected topics. To obtain a broader and more rigorous development of the underlying mathematical concepts than is presented here the interested reader may consult any of a number of standard textbooks and references on statistics and probability.
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- 1.See, for example, L. Lyons, Statistics for nuclear and particle physicists, Cambridge University Press, Cambridge, 1986. A discussion of random variables is given in Chapter 2 of J. Honerkamp, Statistial Physics, Springer-Verlag, New York, 1998.Google Scholar
- 2.In the early days of Monte Carlo it was time consuming to evaluate a logarithm on a computing machine and techniques were developed to permit sampling from the exponential distribution without such an evaluation. For examples see John von Neumann, “Various Techniques Used in Connection With Random Digits,” Monte Carlo Method, A. S. Householder, G. E. Forsythe, and H. H. Germond, eds., National Bureau of Standards Applied Mathematics Series 12, U. S. Government Printing Office, Washington, D.C., 1951, p. 38; and E. D. Cashwell and C. J. Everett, A Practical Manual on the Monte Carlo Methodfor Random Walk Problems, Pergamon Press, New York, 1959, pp. 119–20.Google Scholar
- 3.Robert B. Ash, Real Analysis and Probability, Academic Press, New York, 1972, pp. 321 ff.Google Scholar
- 4.Lyons, op. cit., pp. 13ff.Google Scholar
- 5.This theorem is similar to the “parallel axis theorem” in physics. See Numerical Recipes in Fortran 77: The Art of Scientific Computing, Cambridge University Press, 1986–1992, Chapter 7, p. 308.Google Scholar
- 6.See Reuven Y. Rubinstein, Simulation and The Monte Carlo Method, John Wiley and Sons, New York, 1981, p. 133Google Scholar