# Numerical Quadrature Rules for Common Distributions

## Gauss-Laguerre Quadrature for Exponential Integrals

For integrals of the form ${\int }_{0}^{\infty }{e}^{-\zeta }\varphi \left(\zeta \right)\phantom{\rule{thinmathspace}{0ex}}d\zeta$ for some function $\varphi$, Gauss-Laguerre quadrature of order $n$ provides $n$ abscissæ ${\zeta }_{i}$ and weights ${\omega }_{i}$ for constructing the following linear approximation: ${\int }_{0}^{\infty }{e}^{-\zeta }\varphi \left(\zeta \right)\phantom{\rule{thinmathspace}{0ex}}d\zeta \approx \sum _{i=1}^{n}{\omega }_{i}\varphi \left({\zeta }_{i}\right).$

This form of quadrature is useful for approximating the expectation of a nonlinear function of an exponentially-distributed random variable. Let $X$ be an exponential random variable with rate parameter $\lambda$. Then, with a simple transformation, we can write the expectation of $f\left(X\right)$ in the form above. The expectation of $f\left(X\right)$ is $\mathrm{E}\left[f\left(X\right)\right]={\int }_{0}^{\infty }\lambda {e}^{-\lambda x}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}.$ If we let $\zeta =\lambda x$ and $\varphi \left(\cdot \right)=f\left(\cdot /\lambda \right)$ (and note that $d\zeta =\lambda \phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}$), then, by the change of variables, $\mathrm{E}\left[f\left(X\right)\right]={\int }_{0}^{\infty }{e}^{-\zeta }\varphi \left(\zeta \right)\phantom{\rule{thinmathspace}{0ex}}d\zeta .$ Applying the Gauss-Laguerre quadrature rule above gives the approximation $\mathrm{E}\left[f\left(X\right)\right]\approx \sum _{i=1}^{n}{\omega }_{i}f\left({\zeta }_{i}/\lambda \right).$

For weights and abscissæ, see the Digital Library of Mathematical Functions or the calculator at eFunda.

## Gauss-Hermite Quadrature for Normal Integrals

Similarly, Gauss-Hermite quadrature provides weights ${\omega }_{i}$ and abscissæ ${\zeta }_{i}$ for integral approximations of the form: ${\int }_{-\infty }^{\infty }{e}^{-{\zeta }^{2}}\varphi \left(\zeta \right)\phantom{\rule{thinmathspace}{0ex}}d\zeta \approx \sum _{i=1}^{n}{\omega }_{i}\varphi \left({\zeta }_{i}\right).$ If $X$ is a normally distributed random variable with mean $\mu$ and variance ${\sigma }^{2}$, then we can approximate the expectation of $f\left(X\right)$ using quadrature by applying the transformation $\zeta =\left(x-\mu \right)/\left(\sqrt{2}\sigma \right)$, $\varphi \left(\zeta \right)=f\left(\mu +\sqrt{2}\sigma \zeta \right)/\sqrt{\pi }$, and thus, $d\zeta =\mathrm{dx}/\sqrt{2{\sigma }^{2}}$. Then, $E\left[f\left(X\right)\right]={\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi {\sigma }^{2}}}{e}^{-\frac{\left(x-\mu {\right)}^{2}}{2{\sigma }^{2}}}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}\approx \sum _{i=1}^{n}\frac{{\omega }_{i}}{\sqrt{\pi }}f\left(\mu +\sqrt{2}\sigma {\zeta }_{i}\right).$

For weights and abscissæ, see the Digital Library of Mathematical Functions or the calculator at eFunda.