DISCRETE LAPLACE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.DiscreteLaplace({"a": *, "loc": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PMF, PPF receive float or numpy.ndarray.
distribution.cdf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.pmf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.ppf(int | numpy.ndarray[int]) # -> 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 # -> int
distribution.mode # -> int

Equations

Distribution Definition

$$X\sim \mathrm{DiscreteLaplace}(a,\text{Loc})$$

Distribution Domain

$$x\in \text{Loc}+\mathbb{Z}\equiv \{\dots ,\text{Loc}-1,\text{Loc},\text{Loc}+1,\dots \}$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\{\begin{array}{cl}{\displaystyle \frac{e^{a(x-\text{Loc}+1)}}{1+e^a}}& \text{if }x<\text{Loc}\\ 1-{\displaystyle \frac{e^{-a(x-\text{Loc})}}{1+e^a}}& \text{if }x\ge \text{Loc}\end{array}$$

Probability Density Function

$$f_X\left(x\right)=\tanh \left(\frac{a}{2}\right)e^{-a|x-\text{Loc}|}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\{\begin{array}{cl}\lceil {\displaystyle \frac{\ln \left(u(1+e^a)\right)}{a}}-1\rceil & \text{if }u<\frac{1}{2}\\ \lceil -{\displaystyle \frac{\ln \left((1-u)(1+e^a)\right)}{a}}\rceil & \text{if }u\ge \frac{1}{2}\end{array}$$

Parametric Centered Moments

$$E[X^k]={\mu }_k^{\prime }=\sum _{x\in \text{Loc}+\mathbb{Z}}{x}^k{f}_X\left(x\right)$$

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)=({\mu }_2^{\prime }-{\mu }_1^{\prime 2})={\displaystyle \frac{2e^{-a}}{{(1-e^{-a})}^2}}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=0$$

Parametric Kurtosis

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

Parametric Median

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

Parametric Mode

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

Additional Information and Definitions

• $a:\text{Shape parameter, controls how rapidly the PMF decays from }\text{Loc}$
• $\text{Loc}:\text{Integer location parameter (peak of the distribution)}$
• $u:\text{Uniform[0,1] random varible}$
• $\tanh \left(\frac{a}{2}\right)={\displaystyle \frac{1-e^{-a}}{1+e^{-a}}}$
• $\cosh (a)=\frac{1}{2}(e^a+e^{-a})$
• $\lfloor x\rfloor :\text{Floor function}$
• $\lceil x\rceil :\text{Ceiling Function}$

Spreadsheet Documents

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