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 xi are the roots of the physicists' version of the Hermite polynomial Hn, and the associated weights wi are given byConsider a function h, where the variable y is Normally distributed: . The expectation of h corresponds to the following integral:
As this doesn't exactly correspond to the Hermite polynomial, we need to change variables:
Coupled with the integration by substitution, we obtain:
leading to:
