PearsonDistribution

Pearson probability distribution object

Since R2025a

Description

A PearsonDistribution object consists of parameters and model description for a Pearson probability distribution.

The Pearson distribution is a four-parameter distribution that has an arbitrary mean, standard deviation, skewness, and kurtosis. This distribution is often used to model asymmetric data that is prone to outliers.

The Pearson distribution uses the following parameters:

Parameter	Description
μ	Mean
σ	Standard deviation
γ	Skewness. γ is a measure of the asymmetry of the data around the sample mean. If the skewness is negative, the data spreads out more to the left of the mean than to the right. If the skewness is positive, the data spreads out more to the right. γ2 must be less than κ – 1.
κ	Kurtosis. κ is a measure of how prone a distribution is to outliers. The kurtosis of the normal distribution is 3. Distributions that are more prone to outliers than the normal distribution have a kurtosis value greater than 3; distributions that are less prone have a kurtosis value less than 3. κ must be greater than γ2 + 1.

Creation

Create a Pearson distribution with specified parameter values using the makedist. The default value of this distribution without any specified parameters corresponds to the normal distribution.

Properties

Distribution Parameters

Mu — Mean

0 | scalar

Mean of the Pearson distribution, specified as a scalar.

Data Types: single | double

Sigma — Standard deviation

1 | positive scalar

Standard deviation of the Pearson distribution, specified as a positive scalar.

Data Types: single | double

Gamma — Skewness

0 | scalar

Skewness for the Pearson distribution, specified as a scalar. The value of Gamma must be less than sqrt(Kappa – 1). For more information, see Skewness.

Data Types: single | double

Kappa — Kurtosis

3 | scalar

Kurtosis for the Pearson distribution, specified as a scalar. The value of Kappa must be greater than Gamma^2 + 1. For more information, see Kurtosis.

Data Types: single | double

Distribution Characteristics

IsTruncated — Logical flag for truncated distribution

Read-only: 0 | 1

This property is read-only.

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

Truncation — Truncation interval

Read-only: vector of scalar values

This property is read-only.

Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries.

Data Types: single | double

NumParameters — Number of parameters

Read-only: positive integer value

This property is read-only.

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

ParameterValues — Distribution parameter values

Read-only: vector of scalar values

This property is read-only.

Distribution parameter values, specified as a vector of scalar values.

Data Types: single | double

Type — Type of Pearson distribution

Read-only: integer from interval [0 7]

This property is read-only.

Type of Pearson distribution, specified as an integer from the interval [0 7].

This table describes the distribution corresponding to each Pearson distribution type.

Pearson Distribution Type	Description
0	Normal
1	4-parameter beta
2	Symmetric 4-parameter beta
3	3-parameter gamma
4	Distribution specific to the Pearson system
5	Inverse 3-parameter gamma
6	F location-scale
7	Student's t location-scale

Coefficients — Quadratic polynomial coefficients

Read-only: numeric 1-by-3 vector

This property is read-only.

Quadratic polynomial coefficients, specified as a numeric 1-by-3 vector. The ith element of coefs is the coefficient $$b_i$$ in the differential equation

$$\frac{p\text{'}(x)}{p(x)}=-\frac{a+(x-\mu )}{b_0+b_1(x-\mu )+b_2{(x-\mu )}^2},$$

which defines the Pearson distribution probability density function (pdf) $$p(x)$$.

Support — Support of Pearson distribution

Read-only: numeric 1-by-2 vector

This property is read-only.

The lower and upper support bounds of the Pearson distribution, specified as a numeric 1-by-2 vector.

For more information, see Support.

Other Object Properties

DistributionName — Probability distribution name

Read-only: character vector

This property is read-only.

Probability distribution name, specified as a character vector.

Data Types: char

ParameterDescription — Distribution parameter descriptions

Read-only: cell array of character vectors

This property is read-only.

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

ParameterNames — Distribution parameter names

Read-only: cell array of character vectors

This property is read-only.

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

Object Functions

cdf	Cumulative distribution function
icdf	Inverse cumulative distribution function
iqr	Interquartile range of probability distribution
mean	Mean of probability distribution
median	Median of probability distribution
pdf	Probability density function
plot	Plot probability distribution object
random	Random numbers
std	Standard deviation of probability distribution
truncate	Truncate probability distribution object
var	Variance of probability distribution

Examples

Create Pearson Distribution Object Using Default Parameter Values

Create a Pearson distribution object using the default parameter values, which correspond to the parameters of the standard normal distribution.

pd = makedist("Pearson")

pd = 
  PearsonDistribution

  Pearson distribution
       Mu = 0
    Sigma = 1
    Gamma = 0
    Kappa = 3

You can use the object functions of pd to evaluate the distribution and generate random numbers. Display the supported object functions.

methods(pd)

Methods for class prob.PearsonDistribution:

cdf       gather    icdf      iqr       mean      median    pdf       plot      random    std       truncate  var       

For example, compute the interquartile range of the distribution by using the iqr function.

r = iqr(pd)

r = 
1.3490

Create a Pearson Distribution Object Using Specified Parameters

Create a Pearson distribution object by specifying the parameter values.

pd = makedist("Pearson","mu",5,"sigma",10,"gamma",1,"kappa",4)

pd = 
  PearsonDistribution

  Pearson distribution
       Mu =  5
    Sigma = 10
    Gamma =  1
    Kappa =  4

The type of the distribution is 1, which corresponds to a 4-parameter beta distribution.

pd.Type

ans = 
1

Plot the pdf of the Pearson distribution.

plot(pd)

Version History

Introduced in R2025a