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Pearson system random numbers

`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(...)

`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 + ((

*x*–*a*)/*b*)^{2})^{–c}exp(–*d*arctan((*x*–*a*)/*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

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

Generate random values from the standard normal distribution:

r = pearsrnd(0,1,0,3,100,1); % Equivalent to randn(100,1)

[r,type] = pearsrnd(0,1,1,4,0,0); r = [] type = 1

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