LEVY DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Levy({"mu": *, "c": *})

💡 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

$$X\sim \mathrm{Levy}(\mu ,c)$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=1-\text{erf}\left(\sqrt{\frac{c}{2(x-\mu )}}\right)$$

Probability Density Function

$$f_X\left(x\right)=\sqrt{\frac{c}{2\pi }}\frac{e^{-\frac{c}{2(x-\mu )}}}{(x-\mu {)}^{3/2}}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\frac{c}{2{({\text{erf}}^{-1}(1-u))}^2}$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\mathrm{\infty }$$

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\mu +\frac{c}{2({\text{erf}}^{-1}(1/2){)}^2}$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu +\frac{c}{3}$$

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $c:\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{erf}(x):\text{Error function}$
• ${\mathrm{erf}}^{-1}(x):\text{Inverse of error function}$

Spreadsheet Documents

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