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(x) (i = 1,2,...,n), and the associated weights wi are given by [1] Consider a function h(y), where the variable y is Normally distributed:

The expectation of h corresponds to the following integral:

2 π

exp ⁡

{\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(y-\mu )^{2}}{2\sigma ^{2}}}\right)h(y)dy}

As this does not exactly correspond to the Hermite polynomial, we need to change variables:

σ x + μ

Coupled with the integration by substitution, we obtain:

π

σ x + μ ) d x

leading to:

π