POISSON DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Poisson({"lambda": *})

💡 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

$$X\sim \mathrm{Poisson}\left(\lambda \right)$$

Distribution Domain

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

Parameters Domain and Constraints

$$\lambda \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=e^{-\lambda }\sum _{i=0}^x\frac{{\lambda }^i}{i!}=1-\frac{\gamma (x+1,\lambda )}{x!}=1-P(x-1,\lambda )$$

Probability Density Function

$$f_X\left(x\right)=\frac{{\lambda }^x{e}^{-\lambda }}{x!}$$

Percent Point Function / Sample

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

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\lfloor \lambda +1/3-0.02/\lambda \rfloor$$

Parametric Mode

$$\mathrm{Mode}(X)=\lfloor \lambda \rfloor$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function} \\ \text{is not feasible. However, it is possible to calculate the Percentile Point Function by} \\ \text{approximating it to the nearest integer.} \end{gathered}$$
• $u:\text{Uniform[0,1] random varible}$
• $\lfloor x\rfloor :\text{Floor function}$
• $P(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete Gamma function}$

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