# gauss.quad.prob

##### Gaussian Quadrature with Probability Distributions

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

- Keywords
- math

##### 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 gamma distribution or first shape parameter for beta distribution
- beta
- positive scale parameter for gamma distribution or second shape parameter for beta distribution

##### 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)`

.

##### Value

- A list containing the components
nodes vector of values at which to evaluate the function weights vector of weights to give the function values

##### 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.
Smyth, G. K. (1998). Polynomial approximation. In: *Encyclopedia of Biostatistics*, P. Armitage and T. Colton (eds.), Wiley, London, pp. 3425-3429.
*Gaussian Quadrature Formulas*. Prentice-
Hall, Englewood Cliffs, N.J.

##### See Also

##### 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)
```

*Documentation reproduced from package statmod, version 1.2.4, License: LGPL version 2 or newer*