pearscdf

Pearson cumulative distribution function

Since R2023b

Syntax

f = pearscdf(X,mu,sigma,skew,kurt)

f = pearscdf(X,mu,sigma,skew,kurt,"upper")

[f,type] = pearscdf(___)

[f,type,coefs] = pearscdf(___)

Description

f = pearscdf(X,mu,sigma,skew,kurt) returns the cumulative distribution function (cdf) of the Pearson system evaluated at the values in X, using the mean mu, standard deviation sigma, skewness skew, and kurtosis kurt.

f = pearscdf(X,mu,sigma,skew,kurt,"upper") returns the complement of the Pearson cdf at the values in X. This syntax uses an algorithm that more accurately computes the extreme upper-tail probabilities for all Pearson types except 4.

[f,type] = pearscdf(___) also returns the type of the specified distribution within the Pearson system, using any of the input argument combinations in the previous syntaxes.

[f,type,coefs] = pearscdf(___) also returns the coefficients of the Pearson cdf.

Examples

Compute and Plot Pearson Cumulative Distribution Function

Define the variables mu, sigma, skew, and kurtosis, which contain values for the mean, standard deviation, skewness, and kurtosis of a Pearson distribution, respectively.

mu = 0;
sigma = 2;
skew = 0;
kurtosis = 3;

A Pearson distribution with a skewness of 0 and kurtosis of 3 is equivalent to the normal distribution.

Create a vector X of points from –7 to 7 using the linspace function. Evaluate the cdf for the Pearson distribution given by mu, sigma, skew, and kurtosis at the points in X. Plot the result together with the cdf for the standard normal distribution.

X = linspace(-7,7,1000);
Fp = pearscdf(X,mu,sigma,skew,kurtosis);
Fn = normcdf(X,mu,sigma);

figure
hold on
plot(X,Fp)
plot(X,Fn)
legend(["Pearson CDF" "Normal CDF"])

The plot shows that the blue curve for the Pearson distribution cdf is completely hidden by the red curve for the normal distribution cdf. This result indicates that the Pearson cdf is identical to the normal distribution cdf.

Compute Extreme Upper-Tail Probabilities for Pearson Distribution

Define the variables mu, sigma, skew, and kurtosis, which contain values for the mean, standard deviation, skewness, and kurtosis of a Pearson distribution, respectively.

mu = 2;
sigma = 1;
skew = 2;
kurtosis = 10;

To calculate the probability of sampling a number greater than 150 from the distribution, calculate the probability of sampling a number less than or equal to 150 and subtract it from 1.

p1 = 1 - pearscdf(150,mu,sigma,skew,kurtosis)

p1 = 0

This result shows that the probability of sampling a number less than or equal to 150 is so close to 1 that subtracting it from 1 gives 0.

To approximate the extreme upper-tail probability with greater precision, compute the complement of the Pearson cdf directly.

p2 = pearscdf(150,mu,sigma,skew,kurtosis,"upper")

p2 = 1.0375e-18

The output indicates a small probability of sampling a number greater than 150.

Return Pearson Distribution Type

Define the variables mu, sigma, skew, and kurtosis, which contain values for the mean, standard deviation, skewness, and kurtosis of a Pearson distribution, respectively.

mu = 2;
sigma = 1;
skew = 2;
kurtosis = 10;

Create a vector X of points from 0 to 8 using the linspace function. Evaluate the cdf for the Pearson distribution given by mu, sigma, skew, and kurtosis at the points in X, and return the Pearson distribution type.

X = linspace(0,8,1000);
[f,type] = pearscdf(X,mu,sigma,skew,kurtosis);
type

type = 6

The Pearson distribution defined by mu, sigma, skew, and kurtosis is of type 6. A type 6 Pearson distribution corresponds to the F location scale distribution. In this case, the location is mu and the scale is sigma. For more information, see Pearson Distribution.

Plot the cdf for the type 6 Pearson distribution at the points in X.

plot(X,f)

The plot shows that the cdf evaluates to zero at values less than 0.8. For values greater than 0.8, the cdf increases and then follows the horizontal asymptote corresponding to a probability of 1.

Input Arguments

X — Values at which to evaluate Pearson cdf

scalar | numeric array

Values at which to evaluate the Pearson cdf, specified as a scalar or a numeric array.

To evaluate the cdf at multiple values, specify X using an array. To evaluate the cdfs of multiple distributions, specify either mu or sigma (or both) using arrays. If one or more of the input arguments X, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearscdf expands each scalar input into a constant array of the same size as the array inputs. Each element in f is the cdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in X.

Example: [0 0.4 0.8 0.12]

Data Types: single | double

mu — Mean

scalar | numeric array

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

To evaluate the cdf at multiple values, specify X using an array. To evaluate the cdfs of multiple distributions, specify either mu or sigma (or both) using arrays. If one or more of the input arguments X, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearscdf expands each scalar input into a constant array of the same size as the array inputs. Each element in f is the cdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in X.

Example: [0 1 2; 0 1 2]

Data Types: single | double

sigma — Standard deviation

positive scalar | array of positive values

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

To evaluate the cdf at multiple values, specify X using an array. To evaluate the cdfs of multiple distributions, specify either mu or sigma (or both) using arrays. If one or more of the input arguments X, mu, and sigma are arrays, then the array sizes must be the same. In this case, pearscdf expands each scalar input into a constant array of the same size as the array inputs. Each element in f is the cdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in X.

Example: [1 1 1; 2 2 2]

Data Types: single | double

skew — Skewness

scalar

Skewness for the Pearson distribution, specified as a 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

scalar

Kurtosis for the Pearson distribution, specified as a 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

f — Pearson cdf values

scalar | numeric array

Pearson cdf values evaluated at the values in X returned as a scalar or a numeric array. f is the same size as X, mu, and sigma after any necessary scalar expansion. Each element in f is the cdf value of the distribution specified by skew, kurt, and the corresponding elements in mu and sigma, evaluated at the corresponding value in X.

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, pearscdf 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

numeric 1-by-3 vector

Quadratic polynomial coefficients, returned 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) $$p(x)$$.

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 skewness is negative, the data spreads out more to the left of the mean than to the right. If 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; 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. pearscdf 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

pearscdf is a function specific to the Pearson distribution. Statistics and Machine Learning Toolbox™ also offers the generic function cdf, which supports various probability distributions. To use cdf, specify the probability distribution name and its parameters.

References

[1] Johnson, Norman Lloyd, et al. "Continuous Univariate Distributions." 2nd ed, vol. 1, Wiley, 1994.

[2] Willink, R. "A Closed-Form Expression for the Pearson Type IV Distribution Function." Australian & New Zealand Journal of Statistics, vol. 50, no. 2, June 2008, pp. 199–205. https://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.2008.00508.x.

Extended Capabilities

GPU Arrays

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

Usage notes and limitations:

• For type 4 Pearson distributions, pearscdf takes the same amount of time to run on a GPU as on a CPU.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2023b