A probabilistic model for quadrature rules

The approximate formula of quadrature rules is usually indicated as ∫abf(x)dx≃∑i=1nwif(xi),where xi are the integration nodes and wi are the corresponding coefficients. During last decades, various cases of above type formula, such as Gauss quadrature rules, Newton-Cotes rules and so on, have been introduced to somehow estimate xi and wi. In this paper, we propose a quadrature model based on a probabilistic approach to estimate the related nodes and coefficients. For this purpose, we apply the normal distribution, as a special case, and present an error analysis for the obtained probabilistic formulas. Some numerical examples are given in this way to show the efficiency of the proposed model.