Polynomial Approximations and the Monte-Carlo Method

A new method of computing multiple integrals is proposed, which is a generalization of the ordinary Monte-Caro method. This new method in evaluating the integral makes use of its approximate value as obtained by formulas for mechanical quadratures in accordance with a special distribution law for the integrational points. This new method in evaluating the integral makes use of its approximate value as obtained by formulas for mechanical quadratures in accordance with a special distribution law for the integrational points. It is shown that the standard deviation of the estimation may be considerably decreased, especially when the integrand possesses good differential properties.