ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing various statistical quantities related to the Gamma distribution. More...
Data Types | |
type | distGamma_type |
This is the derived type for signifying distributions that are of type Gamma as defined in the description of pm_distGamma. More... | |
interface | getGammaCDF |
Generate and return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More... | |
interface | getGammaLogPDF |
Generate and return the natural logarithm of the Probability Density Function (PDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More... | |
interface | getGammaLogPDFNF |
Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the Gamma distribution for an input parameter set \((\kappa,\sigma)\). More... | |
interface | setGammaCDF |
Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More... | |
interface | setGammaLogPDF |
Return the natural logarithm of the Probability Density Function (PDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\).More... | |
interface | setGammaRand |
Return a scalar or array of arbitrary rank of Gamma-distributed random values with the specified shape and scale parameters \((\kappa, \sigma)\) of the Gamma distribution corresponding to the procedure arguments (kappa, sigma) . More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distGamma" |
This module contains classes and procedures for computing various statistical quantities related to the Gamma distribution.
Specifically, this module contains routines for computing the following quantities of the Gamma distribution:
The PDF of the Gamma distribution over a strictly-positive support \(x \in (0, +\infty)\) is defined with the two (shape, scale) parameters \((\kappa > 0, \sigma > 0)\) as,
\begin{equation} \large \pi(x | \kappa, \sigma) = \frac {1} {\sigma\Gamma(\kappa)} \bigg(\frac{x}{\sigma}\bigg)^{\kappa - 1} \exp\bigg(-\frac{x}{\sigma}\bigg) ~, \end{equation}
where \(\eta = \frac{1}{\sigma\Gamma(\kappa)}\) is the normalization factor of the distribution with \(\Gamma(\kappa)\) representing the Gamma function whose natural logarithm is returned by the Fortran intrinsic log_gamma()
.
Note that \(\lim_{x\to0} \pi(x | 0 < \kappa < 1, \sigma) \to +\infty\) \(\).
However, this divergence is integrable and the Gamma PDF is properly normalized.
The CDF of the Gamma distribution over a strictly-positive support \(x \in (0, +\infty)\) with the two (shape, scale) parameters \((\kappa > 0, \sigma > 0)\) is defined by the regularized Lower Incomplete Gamma function as,
\begin{eqnarray} \large \mathrm{CDF}(x | \kappa, \sigma) & = & P(\kappa, \frac{x}{\sigma}) \\ & = & \frac{1}{\Gamma(\kappa)} \int_0^{\frac{x}{\sigma}} ~ t^{\kappa - 1}{\mathrm e}^{-t} ~ dt ~, \end{eqnarray}
where \(\Gamma(\kappa)\) represents the Gamma function.
Final Remarks ⛓
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character(*, SK), parameter pm_distGamma::MODULE_NAME = "@pm_distGamma" |
Definition at line 83 of file pm_distGamma.F90.