EXPONENTIAL 2P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Exponential2P({"lambda": *, "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

$$X\sim {\mathrm{Exponential}}_{2\mathrm{P}}(\lambda ,\text{Loc})$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=1-e^{-\lambda (x-\text{Loc})}$$

Probability Density Function

$$f_X\left(x\right)=\lambda e^{-\lambda (x-\text{Loc})}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}-\frac{\ln (1-u)}{\lambda }$$

Parametric Centered Moments

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

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+{\tilde{\mu }}_1^{\prime }=\text{Loc}+\frac{1}{\lambda }$$

Parametric Variance

$$\mathrm{Variance}(X)={\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}=\frac{1}{{\lambda }^2}$$

Parametric Skewness

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

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=9$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\frac{\ln 2}{\lambda }$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{Exponential}\left(\lambda \right)$
• $\text{Loc}:\text{Location parameter}$
• $\lambda :\text{Inverse of scale parameter}$
• $u:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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