MOYAL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Moyal({"mu": *, "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

$$X\sim \mathrm{Moyal}(\mu ,\sigma )$$

Distribution Domain

$$x\in (-\mathrm{\infty },\mathrm{\infty })$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=1-\text{P}(\frac{1}{2},\frac{e^{-z(x)}}{2})=1-\mathrm{erf}\left(\frac{\exp (-0.5z(x))}{\sqrt{2}}\right)$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\sqrt{2\pi }}\exp (-\frac{1}{2}(z(x)+e^{-z(x)}))$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\sigma \ln \left[{\mathrm{\Phi }}^{-1}\left({\left(\frac{1-u}{2}\right)}^2\right)\right]=\mu +\sigma \ln \left[2{\text{P}}^{-1}(\frac{1}{2},1-u)\right]$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\mu +\sigma (\ln (2)+\gamma )$$

Parametric Variance

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

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{28\sqrt{2}\zeta (3)}{{\pi }^3}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=7$$

Parametric Median

$$\mathrm{Median}(X)=\mu +\sigma \ln \left[2{\text{P}}^{-1}(\frac{1}{2},\frac{1}{2})\right]$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu$$

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$
• $\mathrm{erf}(x):\text{Error function}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• $\gamma :\text{Euler-Mascheroni constant}=0.5772156649$
• $\zeta (3):\text{Apéry’s constant}=1.2020569031$

Spreadsheet Documents

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