In numerical analysis, GaussâÂÂHermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:
In this case
where n is the number of sample points used. The x<sub>i</sub> are the roots of the physicists' version of the Hermite polynomial H<sub>n</sub>(x) (i = 1,2,...,n), and the associated weights w<sub>i</sub> are given by
Consider a function h(y), where the variable y is Normally distributed: . The expectation of h corresponds to the following integral:
As this does not exactly correspond to the Hermite polynomial, we need to change variables:
Coupled with the integration by substitution, we obtain:
leading to:
As an illustration, in the simplest non-trivial case, with , we have and , so the estimate reduces to:
â i.e. the average of the function's values one standard deviation below and above the mean.