pm_distCosRaised Module Reference

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"

Detailed Description

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:

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 \(\pi(\cdot)\) 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 \(\ms{Pi}\).

Note

The mean, median, and mode of the distribution correspond to the location parameter \(\mu\).

The variance of the distribution can be computed as,

\begin{equation} \large \ms{variance} = \sigma^2 \left( \frac{1}{3} - \frac{2}{\pi^2} \right) ~. \end{equation}

The distribution has a constant skewness of \(0\).

The distribution has a constant Excess Kurtosis of,

\begin{equation} \large \ms{Ex. Kurtosis} = \frac{6(90 - \pi^{4})}{5(\pi^{2} - 6)^{2}} \approx -0.59376 ~. \end{equation}

See also

pm_distUnif

Test:

test_pm_distCosRaised

Todo:

Normal Priority: Two additional interfaces for computing the quantiles and random values of Raised Cosine Distribution must be added.
The methodology employed for the [Beta distribution](@ pm_distBeta) might be useful 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_distCosRaised::MODULE_NAME = "@pm_distCosRaised"

Definition at line 89 of file pm_distCosRaised.F90.