Monte Carlo simulations

An important part of statistical analysis is to be able to evaluate expectations of random variables. However, under certain circumstances, it is not feasible to apply a deterministic algorithm or difficult to obtain a closed-form expression. In other words, it is difficult or impossible to express the relationship between the explanatory and response variables under analytical terms using a finite number of elementary functions such as constants, exponents, n roots, and logarithms. A practical way to solve these problems is to use Monte Carlo methods, which are a broad class of computational algorithms that rely on repeated random sampling of quantities to approximate the distribution of an unknown probability distribution.

Methods based on Monte Carlo are used to approximate the properties of random variables. For example, to estimate mean = E(X) of a distribution using the Monte Carlo method, we generate m independent and identically distributed copies of X, namely...