BERNOULLI DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Bernoulli({"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

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

Distribution Domain

$$x\in \{0,1\}$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\{\begin{array}{cl}1-p& \text{if }x=0\\ 1& \text{if }x=1\end{array}$$

Probability Density Function

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

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\{\begin{array}{cl}1& \text{if }u\le p\\ 0& \text{if }u>p\end{array}$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\{\begin{array}{cl}0& \text{if }p<1/2\\ [0,1]& \text{if }p=1/2\\ 1& \text{if }p>1/2\end{array}$$

Parametric Mode

$$\mathrm{Mode}(X)=\{\begin{array}{cl}0& \text{if }p<1/2\\ 0,1& \text{if }p=1/2\\ 1& \text{if }p>1/2\end{array}$$

Additional Information and Definitions

• $u:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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