BINOMIAL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Binomial({"n": *, "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

$$X\sim \mathrm{Binomial}(n,p)$$

Distribution Domain

$$x\in \mathbb{N}\equiv \{0,1,2,\dots \}$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\sum _{i=0}^x(\genfrac{}{}{0ex}{}{n}{i})p^i(1-p{)}^{n-i}=I(1-p,n-x,1+x)$$

Probability Density Function

$$f_X\left(x\right)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}$$

Percent Point Function / Sample

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

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\lfloor np\rfloor \vee \lceil np\rceil$$

Parametric Mode

$$\mathrm{Mode}(X)=\lfloor (n+1)p\rfloor \vee \lceil (n+1)p\rceil -1$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function} \\ \text{is not feasible. However, it is possible to calculate the Percentile Point Function by} \\ \text{approximating it to the nearest integer.} \end{gathered}$$
• $u:\text{Uniform[0,1] random varible}$
• $\lfloor x\rfloor :\text{Floor function}$
• $\lceil x\rceil :\text{Ceiling Function}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $S(a,b):\text{Stirling numbers of the second kind}=\frac{1}{b!}\sum _{j=0}^b(-1{)}^{b-j}(\genfrac{}{}{0ex}{}{b}{j})j^a$

Spreadsheet Documents

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