GUMBEL LEFT DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GumbelLeft({"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

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

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

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

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\sigma }\exp (z(x)-e^{z(x)})$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\sigma \ln (-\ln (1-u))$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)={\sigma }^2({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\sigma }^2\frac{{\pi }^2}{6}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=-\frac{12\sqrt{6}\zeta (3)}{{\pi }^3}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=3+\frac{12}{5}$$

Parametric Median

$$\mathrm{Median}(X)=\mu +\sigma \ln (-\ln \left(\frac{1}{2}\right))$$

Parametric Mode

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

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{GumbelLeft}(0,1)$
• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\gamma :\text{Euler-Mascheroni constant}=0.5772156649$
• $\zeta (3):\text{Apéry’s constant}=1.2020569031$

Spreadsheet Documents

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