# pearsrnd

Pearson system random numbers

## Syntax

```r = pearsrnd(mu,sigma,skew,kurt,m,n) r = pearsrnd(mu,sigma,skew,kurt) r = pearsrnd(mu,sigma,skew,kurt,m,n,...) r = pearsrnd(mu,sigma,skew,kurt,[m,n,...]) [r,type] = pearsrnd(...) [r,type,coefs] = pearsrnd(...) ```

## Description

`r = pearsrnd(mu,sigma,skew,kurt,m,n)` returns an `m`-by-`n` matrix of random numbers drawn from the distribution in the Pearson system with mean `mu`, standard deviation `sigma`, skewness `skew`, and kurtosis `kurt`. The parameters `mu`, `sigma`, `skew`, and `kurt` must be scalars.

### Note

Because `r` is a random sample, its sample moments, especially the skewness and kurtosis, typically differ somewhat from the specified distribution moments.

`pearsrnd` uses the definition of kurtosis for which a normal distribution has a kurtosis of 3. Some definitions of kurtosis subtract 3, so that a normal distribution has a kurtosis of 0. The `pearsrnd` function does not use this convention.

Some combinations of moments are not valid; in particular, the kurtosis must be greater than the square of the skewness plus 1. The kurtosis of the normal distribution is defined to be 3.

`r = pearsrnd(mu,sigma,skew,kurt)` returns a scalar value.

`r = pearsrnd(mu,sigma,skew,kurt,m,n,...)` or ```r = pearsrnd(mu,sigma,skew,kurt,[m,n,...])``` returns an `m`-by-`n`-by-... array.

`[r,type] = pearsrnd(...)` returns the type of the specified distribution within the Pearson system. `type` is a scalar integer from `0` to `7`. Set `m` and `n` to `0` to identify the distribution type without generating any random values.

The seven distribution types in the Pearson system correspond to the following distributions:

• `0`Normal distribution

• `1` — Four-parameter beta distribution

• `2` — Symmetric four-parameter beta distribution

• `3` — Three-parameter gamma distribution

• `4` — Not related to any standard distribution. The density is proportional to:

(1 + ((xa)/b)2)c exp(–d arctan((xa)/b)).

• `5` — Inverse gamma location-scale distribution

• `6`F location-scale distribution

• `7` — Student's t location-scale distribution

`[r,type,coefs] = pearsrnd(...)` returns the coefficients `coefs` of the quadratic polynomial that defines the distribution via the differential equation

`$\frac{d}{dx}\mathrm{log}\left(p\left(x\right)\right)=\frac{-\left(a+x\right)}{c\left(0\right)+c\left(1\right)x+c\left(2\right){x}^{2}}.$`

## Examples

Generate random values from the standard normal distribution:

`r = pearsrnd(0,1,0,3,100,1); % Equivalent to randn(100,1)`
Determine the distribution type:
```[r,type] = pearsrnd(0,1,1,4,0,0); r = [] type = 1```

## References

 Johnson, N.L., S. Kotz, and N. Balakrishnan (1994) Continuous Univariate Distributions, Volume 1, Wiley-Interscience, Pg 15, Eqn 12.33.