Any method which solves a problem by generating suitable random numbers and observing that fraction of the numbers obeying some property or properties. The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically. It was named by S. Ulam , who in 1946 became the first mathematician to dignify this approach with a name, in honor of a relative having a propensity to gamble (Hoffman 1998, p. 239). Nicolas Metropolis also made important contributions to the development of such methods.
The most common application of the Monte Carlo method is Monte Carlo integration.
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In order to integrate a function over a complicated domain , Monte Carlo integration picks random points over some simple domain which is a superset of , checks whether each point is within , and estimates the area of (volume, -dimensional content, etc.) as the area of multiplied by the fraction of points falling within . Monte Carlo integration is implemented in Mathematica as NIntegrate[f, ..., Method->MonteCarlo].
Picking randomly distributed points , , ..., in a multidimensional volume to determine the integral of a function in this volume gives a result
(Press et al. 1992, p. 295).
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