Monte carlo integrations and variance reduction techniques for N-Dimensional Integrals

In this work, we consider two Monte Carlo methods for evaluating the n-dimensional integrals for bounded integrand. Statistical properties of these methods are illustrated and unified. The supported number of trials to estimate the integrals, confidence interval and the efficiency for each method were derived theoretically and assessed practically. Variance Reduction for Monte Carlo methods is discussed theoretically and explained by algorithms where four techniques are considers, namely, the Importance Sampling, the Correlated Sampling, the Partition of the region, and the Biased Estimator. The computer programs are illustrated in appendices by the run is made by using MathCAD 2001i.