# gauss.quad.prob: Gaussian Quadrature with Probability Distributions

## Description

Calculate nodes and weights for Gaussian quadrature in terms of probability distributions.## Usage

gauss.quad.prob(n,dist="uniform",l=0,u=1,mu=0,sigma=1,alpha=1,beta=1)

## Arguments

n

number of nodes and weights

dist

distribution that Gaussian quadrature is based on, one of `"uniform"`

, `"normal"`

, `"beta"`

or `"gamma"`

l

lower limit of uniform distribution

u

upper limit of uniform distribution

mu

mean of normal distribution

sigma

standard deviation of normal distribution

alpha

positive shape parameter for beta or gamma distribution

beta

positive scale parameter for gamma distribution

## Value

- A list containing the components
- nodesvector of values at which to evaluate the function
- weightsvector of weights to give the function values

## Details

This is a rewriting and simplification of `gauss.quad`

in terms of probability distributions.
The expected value of `f(X)`

is approximated by `sum(w*f(x))`

where `x`

is the vector of nodes and `w`

is the vector of weights. The approximation is exact if `f(x)`

is a polynomial of order no more than `2n+1`

.
The possible choices for the distribution of `X`

are as follows:
Uniform on `(l,u)`

.
Normal with mean `mu`

and standard deviation `sigma`

.
Beta with density `x^(alpha-1)*(1-x)^(beta-1)/B(alpha,beta)`

on `(0,1)`

.
Gamma with density `x^(alpha-1)*exp(-x/beta)/beta^alpha/gamma(alpha)`

.## References

Golub, G. H., and Welsch, J. H. (1969). Calculation of Gaussian
quadrature rules. *Mathematics of Computation* **23**, 221-230.
Golub, G. H. (1973). Some modified matrix eigenvalue problems.
*Siam Review* **15**, 318-334.
Stroud and Secrest (1966). *Gaussian Quadrature Formulas*. Prentice-
Hall, Englewood Cliffs, N.J.## Examples

out <- gauss.quad.prob(10,"normal")
sum(out$weights * out$nodes^4)
# the 4th moment of the standard normal is 3
out <- gauss.quad.prob(32,"gamma",alpha=5)
sum(out$weights * log(out$nodes))
# the expected value of log(X) where X is gamma is digamma(alpha)