MAXWELL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Maxwell({"alpha": *, "loc": *})

💡 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

$$X\sim \mathrm{Maxwell}(\alpha ,\text{Loc})$$

Distribution Domain

$$x\in (0,\mathrm{\infty })$$

Parameters Domain and Constraints

$$\alpha \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\mathrm{erf}\left(\frac{x-\text{Loc}}{\sqrt{2}\alpha }\right)-\sqrt{\frac{2}{\pi }}\frac{(x-\text{Loc})e^{-(x-\text{Loc}{)}^2/\left(2{\alpha }^2\right)}}{\alpha }$$

Probability Density Function

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

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\alpha \sqrt{2{\text{P}}^{-1}(1.5,u)}$$

Parametric Centered Moments

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

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\text{Loc}+2\alpha \sqrt{\frac{2}{\pi }}$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{{\alpha }^2(3\pi -8)}{\pi }$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2\sqrt{2}(16-5\pi )}{(3\pi -8{)}^{3/2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=4\frac{(-96+40\pi -3{\pi }^2)}{(3\pi -8{)}^2}+3$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\alpha \sqrt{2{\text{P}}^{-1}(1.5,\frac{1}{2})}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\alpha \sqrt{2}$$

Additional Information and Definitions

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

Spreadsheet Documents

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