pearsinv

Pearson inverse cumulative distribution function (icdf)

Since R2025a

Syntax

x = pearsinv(p,mu,sigma,skew,kurt)

[x,type] = pearsinv(p,mu,sigma,skew,kurt)

[x,type,coefs] = pearsinv(p,mu,sigma,skew,kurt)

Description

x = pearsinv(p,mu,sigma,skew,kurt) returns the inverse of the Pearson cumulative distribution function (icdf), evaluated at the probability values in p, using the mean mu, standard deviation sigma, skewness skew, and kurtosis kurt.

[x,type] = pearsinv(p,mu,sigma,skew,kurt) also returns the type of the specified distribution within the Pearson system.

[x,type,coefs] = pearsinv(p,mu,sigma,skew,kurt) also returns the coefficients of the Pearson pdf.

Examples

Inverse of Pearson Distribution cdf

Compute the inverse of cdf (icdf) values evaluated at the probability values in p for the Pearson distribution with mean mu, standard deviation sigma, skewness skew, and kurtosis kurt.

p = 0.25:0.25:1;
mu = 2;
sigma = 1;
skew = 1;
kurt = 4;
x = pearsinv(p,mu,sigma,skew,kurt)

x = 1×4

    1.2345    1.8045    2.5661   10.5249

Compute the cdf at x using the pearscdf function with the same input argument values.

p2 = pearscdf(x,mu,sigma,skew,kurt)

p2 = 1×4

    0.2500    0.5000    0.7500    1.0000

p2-p 

ans = 1×4
10-14 ×

    0.0056   -0.0056   -0.2220         0

The values in p2 and p are the same, with small floating point numerical differences. So, pearscdf and pearsinv are complements.

Compute the icdf values evaluated at 0.5 for various Pearson distributions with different mean values.

mu = [-2,-1,0,1,2];
[x,type,coefs] = pearsinv(0.5,mu,sigma,skew,kurt)

x = 1×5

   -2.1955   -1.1955   -0.1955    0.8045    1.8045

type = 
1

coefs = 1×3

    1.3000    0.7000   -0.1000

The output displays the coefficients for the distribution pdf and indicates that the Pearson distribution is type 1. A type 1 Pearson distribution corresponds to the 4-parameter beta distribution. For more information, see Pearson Distribution.

Input Arguments

p — Probability values at which to evaluate inverse of cdf

numeric scalar in [0,1] | numeric array

Probability values at which to evaluate the inverse of the cdf (icdf), specified as a numeric scalar or a numeric array, where each element is in the range [0,1].

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearsinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [0.1,0.5,0.9]

Data Types: single | double

mu — Mean

numeric scalar | numeric array

Mean of the Pearson distribution, specified as a numeric scalar or a numeric array.

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearsinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [0 1 2; 0 1 2]

Data Types: single | double

sigma — Standard deviation

positive numeric scalar | positive numeric array

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

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearsinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [1 1 1; 2 2 2]

Data Types: single | double

skew — Skewness

numeric scalar

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

Example: 3

Data Types: single | double

kurt — Kurtosis

numeric scalar

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

Example: 11

Data Types: single | double

Output Arguments

x — icdf values

numeric scalar | numeric array

icdf values, evaluated at the probability values in p, returned as a numeric scalar or a numeric array. x is the same size as p, mu, and sigma after any necessary scalar expansion. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p. The distribution has the specified scalar values skew and kurt.

type — Type of Pearson distribution

integer from interval [0 7] | NaN

Type of Pearson distribution used to calculate the cdf, returned as an integer from the interval [0 7] or NaN. If the distribution parameters are invalid, pearsinv returns NaN.

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

coefs — Quadratic polynomial coefficients

1-by-3 numeric vector

Quadratic polynomial coefficients, returned as a 1-by-3 numeric 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) $$p(x)$$.

You can calculate the support for the Pearson distribution cdf using coefs. For more information, see Probability Density Function and Support.

More About

Skewness

Skewness 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. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero.

The skewness of a distribution is defined as

$$s=\frac{E{\left(x-\mu \right)}^3}{{\sigma }^3},$$

where µ is the mean of x, σ is the standard deviation of x, and E(t) represents the expected value of the quantity t.

Kurtosis

Kurtosis 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, and distributions that are less prone have a kurtosis value less than 3. Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has a kurtosis of 0. pearsinv does not use this convention.

The kurtosis of a distribution is defined as

$$k=\frac{E{(x-\mu )}^4}{{\sigma }^4},$$

where μ is the mean of x, σ is the standard deviation of x, and E(t) represents the expected value of the quantity t.

Alternative Functionality

• pearsinv is a function specific to the Pearson distribution. Statistics and Machine Learning Toolbox™ also offers the generic function icdf, which supports various probability distributions. To use icdf, create a PearsonDistribution probability distribution object and pass the object as an input argument to icdf. Alternatively, you can specify the probability distribution name and parameters in the call to icdf. Note that the distribution-specific function pearsinv is faster than the generic function icdf.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2025a