WEIBULL 3P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Weibull3P({"alpha": *, "loc": *, "beta": *})

💡 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

$$X\sim {\mathrm{Weibull}}_{3\mathrm{P}}(\alpha ,\text{Loc},\beta )$$

Distribution Domain

$$x\in [\text{Loc},\mathrm{\infty })$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

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

Probability Density Function

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

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\beta (-\ln (1-u){)}^{1/\alpha }$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx={\beta }^{\alpha }\mathrm{\Gamma }(1+\frac{k}{\alpha })$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+{\tilde{\mu }}_1^{\prime }=\text{Loc}+\beta \mathrm{\Gamma }(1+1/\alpha )$$

Parametric Variance

$$\mathrm{Variance}(X)={\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}={\beta }^2[\mathrm{\Gamma }(1+2/\alpha )-{\left(\mathrm{\Gamma }(1+1/\alpha )\right)}^2]$$

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\beta (\ln (2){)}^{1/\alpha }$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\{\begin{array}{cl}\beta {\left(\frac{\alpha -1}{\alpha }\right)}^{1/\alpha }& \text{if }\alpha >1\\ 0& \text{if }\alpha \le 1\end{array}$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{Weibull}(\alpha ,\beta )$
• $\text{Loc}:\text{Location parameter}$
• $\beta :\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\beta$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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