By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution.

When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. the 'sample mean') of independent samples of the variable. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean ( a.k.a. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative 'soft' methods. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. The underlying concept is to use randomness to solve problems that might be deterministic in principle. Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.

Not to be confused with Monte Carlo algorithm.

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