ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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This module contains classes and procedures for computing various statistical quantities related to the Raised Cosine distribution. More...
Data Types | |
type | distCosRaised_type |
This is the derived type for signifying distributions that are of type Raised Cosine as defined in the description of pm_distCosRaised. More... | |
interface | getCosRaisedCDF |
Generate and return the Cumulative Distribution Function (CDF) of the Raised Cosine distribution for an input x within the support of the distribution \(x \in [\mu - \sigma, \mu + \sigma]\). More... | |
interface | getCosRaisedPDF |
Generate and return the Probability Density Function (PDF) of the Raised Cosine distribution for an input x within the support of the distribution \(x \in [\mu - \sigma, \mu + \sigma]\). More... | |
interface | setCosRaisedCDF |
Return the Cumulative Distribution Function (CDF) of the Raised Cosine distribution for an input x within the support of the distribution \(x \in [\mu - \sigma, \mu + \sigma]\). More... | |
interface | setCosRaisedPDF |
Return the Probability Density Function (PDF) of the Raised Cosine distribution for an input x within the support of the distribution \(x \in [\mu - \sigma, \mu + \sigma]\). More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distCosRaised" |
This module contains classes and procedures for computing various statistical quantities related to the Raised Cosine distribution.
Specifically, this module contains routines for computing the following quantities of the Raised Cosine distribution:
The PDF \(\pi(\cdot)\) of the Raised Cosine distribution over a strictly-positive support \(x \in [\mu - \sigma, \mu + \sigma]\) is defined with the two (location, scale) parameters \((\mu, \sigma)\) as,
\begin{equation} \large \pi(x | \mu, \sigma) = \frac{1}{2\sigma} \left[ 1 + \cos \left({\frac {x - \mu}{\sigma}} ~ \pi \right) \right] ~,~ x \in [\mu - \sigma, \mu + \sigma] \end{equation}
where \(\pi\) on the right hand side expression is the number \(\ms{Pi}\).
The CDF \(\ms{CDF}(\cdot)\) of the Raised Cosine distribution over a strictly-positive support \(x \in [\mu - \sigma, \mu + \sigma]\) is defined with the two (location, scale) parameters \((\mu, \sigma)\) as,
\begin{equation} \large \ms{CDF}(x | \mu, \sigma) = \frac {1}{2} \left[ 1 + \frac{x - \mu}{\sigma} + \frac{1}{\pi} \sin\left( \frac{x - \mu}{\sigma} \pi \right) \right] ~,~ x \in [\mu - \sigma, \mu + \sigma] \end{equation}
where \(\pi\) on the right hand side expression is the number \(\ms{Pi}\).
\begin{equation} \large \ms{variance} = \sigma^2 \left( \frac{1}{3} - \frac{2}{\pi^2} \right) ~. \end{equation}
\begin{equation} \large \ms{Ex. Kurtosis} = \frac{6(90 - \pi^{4})}{5(\pi^{2} - 6)^{2}} \approx -0.59376 ~. \end{equation}
Final Remarks ⛓
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character(*, SK), parameter pm_distCosRaised::MODULE_NAME = "@pm_distCosRaised" |
Definition at line 89 of file pm_distCosRaised.F90.