GENERALIZED NORMAL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GeneralizedNormal({"beta": *, "mu": *, "alpha": *})

💡 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

$$X\sim \mathrm{GeneralizedNormal}(\beta ,\mu ,\alpha )$$

Distribution Domain

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

Parameters Domain and Constraints

$$\beta \in {\mathbb{R}}^{+},\mu \in \mathbb{R},\alpha \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{2}+\frac{\text{sign}(x-\mu )}{2\mathrm{\Gamma }(1/\beta )}\gamma (1/\beta ,{\left|\frac{x-\mu }{\alpha }\right|}^{\beta })=\frac{1}{2}+\frac{\text{sign}(x-\mu )}{2}\text{P}(1/\beta ,{\left|\frac{x-\mu }{\alpha }\right|}^{\beta })$$

Probability Density Function

$$f_X\left(x\right)=\frac{\beta }{2\alpha \mathrm{\Gamma }(1/\beta )}\exp (-{\left(\frac{|x-\mu |}{\alpha }\right)}^{\beta })$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{sign}(u-\frac{1}{2}){\left[{\alpha }^{\beta }{\text{P}}^{-1}(\frac{1}{\beta },2|u-\frac{1}{2}|)\right]}^{1/\beta }+\mu$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=\{\begin{array}{cl}0& \text{if }k\text{ is odd}\\ {\alpha }^k\mathrm{\Gamma }\left(\frac{k+1}{\beta }\right)/\mathrm{\Gamma }\left(\frac{1}{\beta }\right)& \text{if }k\text{ is even}\end{array}$$

Parametric Mean

$$\mathrm{Mean}(X)=\mu +\alpha {\mu }_1^{\prime }=\mu$$

Parametric Variance

$$\mathrm{Variance}(X)={\alpha }^2({\mu }_2^{\prime }-{\mu }_1^{\prime 2})=\frac{{\alpha }^2\mathrm{\Gamma }(3/\beta )}{\mathrm{\Gamma }(1/\beta )}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=0$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{\mathrm{\Gamma }(5/\beta )\mathrm{\Gamma }(1/\beta )}{\mathrm{\Gamma }(3/\beta {)}^2}$$

Parametric Median

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

Parametric Mode

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

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $\alpha :\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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