Gauss–Hermite quadrature

In numerical analysis, Gauss–Hermite 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_i are the roots of the physicists' version of the Hermite polynomial H_n, and the associated weights w_i are given by

Example with change of variable

Consider a function h, 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:
– i.e. the average of the function's values one standard deviation below and above the mean.