GEOMETRIC DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Geometric({"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

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

Distribution Domain

$$x\in {\mathbb{N}}^{+}\equiv \{1,2,\dots \}$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=1-(1-p{)}^{\lfloor x\rfloor }$$

Probability Density Function

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

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\lceil \frac{\ln (1-u)}{\ln (1-p)}\rceil$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2-p}{\sqrt{1-p}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=9+\frac{p^2}{1-p}$$

Parametric Median

$$\mathrm{Median}(X)=\lceil \frac{-1}{{\log }_2(1-p)}\rceil$$

Parametric Mode

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

Additional Information and Definitions

• $u:\text{Uniform[0,1] random varible}$
• $\lfloor x\rfloor :\text{Floor function}$
• $\lceil x\rceil :\text{Ceiling Function}$

Spreadsheet Documents

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