RAYLEIGH DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Rayleigh({"gamma": *, "sigma": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{Rayleigh}(\gamma ,\sigma )$$

Distribution Domain

$$x\in [\gamma ,\mathrm{\infty })$$

Parameters Domain and Constraints

$$\gamma \in \mathbb{R},\sigma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-e^{-z(x{)}^2/2}$$

Probability Density Function

$$f_X\left(x\right)=z(x)\times e^{-z(x{)}^2/2}/\sigma$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\gamma +\sigma \sqrt{-2\log (1-u)}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\sqrt{2^k}\mathrm{\Gamma }(\frac{k}{2}+1)$$

Parametric Mean

$$\mathrm{Mean}(X)=\gamma +\sigma \cdot {\mu }_1^{\prime }=\gamma +\sigma \sqrt{\frac{\pi }{2}}$$

Parametric Variance

$$\mathrm{Variance}(X)={\sigma }^2({\mu }_2^{\prime }-{\mu }_1^{\prime 2})={\sigma }^2\frac{4-\pi }{2}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2(\pi -3)\sqrt{\pi }}{{(4-\pi )}^{3/2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=3+\frac{24\pi -6{\pi }^2-16}{{(4-\pi )}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\gamma +\sigma \sqrt{-2\log \left(\frac{1}{2}\right)}$$

Parametric Mode

$$\mathrm{Mode}(X)=\gamma +\sigma$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{Rayleigh}(0,1)$
• $\gamma :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\gamma )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document