LOGARITHMIC DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Logarithmic({"p": *})

💡 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

$$X\sim \mathrm{Logarithmic}\left(p\right)$$

Distribution Domain

$$x\in {\mathbb{N}}_{⩾1}\equiv \{1,2,\dots \}$$

Parameters Domain and Constraints

$$p\in (0,1)\subseteq \mathbb{R}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\sum _{i=0}^x\frac{1}{-\ln (1-p)}\frac{p^i}{i}$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{-\ln (1-p)}\frac{p^x}{x}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\arg \underset{x}{min}|F_X\left(x\right)-u|$$

Parametric Centered Moments

$$E[X^k]={\mu }_k^{\prime }=\sum _{x=0}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)=\frac{(k-1)!}{-\ln (1-p)}{\left(\frac{p}{1-p}\right)}^k$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\frac{1}{-\ln (1-p)}\frac{p}{1-p}$$

Parametric Variance

$$\mathrm{Variance}(X)=({\mu }_2^{\prime }-{\mu }_1^{\prime 2})=-\frac{p^2+p\ln (1-p)}{(1-p{)}^2(\ln (1-p){)}^2}$$

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=F_X^{-1}\left(0.5\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=1$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function} \\ \text{is not feasible. However, it is possible to calculate the Percentile Point Function by} \\ \text{approximating it to the nearest integer.} \end{gathered}$$
• $u:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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