Monte Carlo Methods

Pseudo Random Number Generators

Initially, the basis of the Monte Carlo method was the use of uniform pseudo random numbers on the interval [0,1]. Today, Monte Carlo methods (plural) apply to the use of any pseudo random number generator, such as variates obtained from the exponential distribution, or repeated sampling from large, real data sets.

Programming Environment

Since its commercial release by the IBM Corporation in 1957, FORTRAN (FORmula TRANslator) remains the fastest high-level programming language for Monte Carlo simulation work. This is because execution time is an essential component for realistic, applied problems. Higher level programming environments, such as SAS IML, S+, and Lucent Technology's R (which is available at no cost from http://www.rproject.org/), are serviceable for simple Monte Carlo simulations and classroom demonstrations.

Simulating Tossing a Die

A Monte Carlo computer solution easily simulates the tossing of a fair die with a variate drawn from a uniform [0,1] pseudo random number generator and a table of assigned outcomes, such as indicated in Table 1. For example, if the value obtained from the generator is .1770, it simulates the throwing of a fair die and obtaining two spots. Counting the results of repeating this simulation is tremendously faster and more accurate than physically tossing a die.

Estimating the Area of a Regular Figure

Consider the area (A) of a well-defined geometric shape, such as that in the shaded area depicted in Figure 1. The area of interest is bounded by the two equations f(x) = x and g(x) = x2 over the interval (0, 1).

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