pm_distBeta Module Reference

This module contains classes and procedures for computing various statistical quantities related to the Beta distribution. More...

Data Types
type	distBeta_type
	This is the derived type for signifying distributions that are of type Beta as defined in the description of pm_distBeta. More...
interface	getBetaCDF
	Generate and return the CDF of the Beta distribution for the given parameters \((\alpha, \beta)\) as defined in the details section of pm_distBeta. More...
interface	getBetaLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the Beta distribution for an input x within the support of the distribution \(x \in (0,1)\). More...
interface	getBetaPDF
	Generate and return the Probability Density Function (PDF) of the Beta distribution for an input x within the support of the distribution \(x \in [0,1]\). More...
interface	setBetaCDF
	Return the CDF of the Beta distribution for the given parameters \((\alpha, \beta)\) as defined in the details section of pm_distBeta. More...
interface	setBetaLogPDF
	Return the natural logarithm of the Probability Density Function (PDF) of the Beta distribution for an input x within the support of the distribution \(x \in (0,1)\). More...
interface	setBetaRand
	Return a scalar or array of arbitrary rank of Beta-distributed random values in range \([0, 1]\) (or \((0, 1)\), depending on the specific parameter values) with the specified two shape parameters \((\alpha, \beta)\) of the Beta distribution corresponding to the procedure arguments (alpha, beta). More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distBeta"

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the Beta distribution.

Specifically, this module contains routines for computing the following quantities of the Beta distribution:

1. the Probability Density Function (PDF)
2. the Cumulative Distribution Function (CDF)
3. the Random Number Generation from the distribution (RNG)
4. the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

The PDF of the Beta distribution is defined with two shape parameters \((\alpha > 0, \beta > 0)\) as,

\begin{equation} \large \pi(x | \alpha, \beta) = \begin{cases} \frac {x^{\alpha - 1} (1 - x)^{\beta - 1}} {\mathrm{B}(\alpha, \beta)} &,~ x \in (0, 1) ~, \\ 0 &,~ x \notin (0,1) ~. \end{cases} \end{equation}

where,

\begin{equation} \mathrm{B}(\alpha, \beta) = \frac {\Gamma(\alpha) \Gamma(\beta)} {\Gamma(\alpha + \beta)} \end{equation}

is the Beta function whose natural logarithm is returned by getLogBeta.
Note that the common definitions of the Beta distribution consider the end points \( x = 0, x = 1\) also as part of the support of the distribution.
But this inclusion is valid only if the condition \((\alpha \geq 1, \beta \geq 1)\) holds.
Furthermore, the end points correspond to a PDF value of either \(0\) or \(+\infty\), yielding a NaN when the natural logarithm of the PDF is returned in numerical algorithms.
As such, the support of the Beta distribution in this module is always taken to be the open-interval \(x \in (0,1)\).

The CDF of the Beta distribution is exactly given by the regularized incomplete Beta function.
See the documentation of pm_mathBeta for details of the CDF of the Beta distribution.

Random Number Generation

If \(X\) and \(Y\) are independent, with \(X \sim \Gamma(\alpha, \theta)\) and \(Y \sim \Gamma(\beta, \theta)\) then,

\begin{equation} \frac{X}{X + Y} \sim \mathrm{B}(\alpha, \beta) ~, \end{equation}

where \((\alpha, \beta)\) are the parameters of the target Beta distribution and, \(\theta\) is an arbitrary parameter of the Gamma distribution.
So one algorithm for generating random Beta variates is to generate \(\frac{X}{X + Y}\), where \(X\) is a Gamma variate with parameters \((\alpha, 1)\) and, \(Y\) is an independent gamma variate with parameters \((\beta, 1)\).

Note

When \(\alpha > 1\), the PDF of the Beta distribution is bound to zero at \(x = 0\).
When \(\beta > 1\), the PDF of the Beta distribution is bound to zero at \(x = 1\).
When \(\alpha \leq 1\), the PDF of the Beta distribution is unbounded ( \(+\infty\)) at \(x = 0\).
When \(\beta \leq 1\), the PDF of the Beta distribution is unbounded ( \(+\infty\)) at \(x = 1\).

The Cumulative Distribution Function (CDF) of the Beta distribution for a triple \((x, \alpha, \beta)\) is directly returned by calling getBetaInc(x, alpha, beta).
Similarly, the Cumulative Distribution Function (CDF) of the Beta distribution for a triple \((x, \alpha, \beta)\) is directly returned by calling getBetaInv(x, alpha, beta).

See also

pm_mathBeta
pm_distUnif
pm_distGamma
pm_mathGamma

Test:

test_pm_distBeta

Todo:

Critical Priority: The quantile function must be implemented here.

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_distBeta::MODULE_NAME = "@pm_distBeta"

Definition at line 107 of file pm_distBeta.F90.