SKELLAM DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.discrete.Skellam({"lambda1": *, "lambda2": *})

💡 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

$$X\sim \mathrm{Skellam}({\lambda }_1,{\lambda }_2)$$

Distribution Domain

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

Parameters Domain and Constraints

$${\lambda }_1\in {\mathbb{R}}^{+},{\lambda }_2\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\sum _{k=-\mathrm{\infty }}^x{f}_X\left(k\right)$$

Probability Density Function

$$f_X\left(x\right)=e^{-({\lambda }_1+{\lambda }_2)}{\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{x/2}{I}_{\left|x\right|}\left(2\sqrt{{\lambda }_1{\lambda }_2}\right)$$

Percent Point Function / Sample

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

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

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

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{k\in \{\lfloor {\lambda }_1-{\lambda }_2\rfloor ,\lceil {\lambda }_1-{\lambda }_2\rceil \}}{max}{f}_X(k)$$

Additional Information and Definitions

• $\text{Computing an analytic expression for the inverse of the cumulative distribution function}$ $\text{is not feasible. However, it is possible to calculate the Percentile Point Function by}$ $\text{approximating it to the nearest integer.}$
• $X=N_1-N_2,N_1\sim \mathrm{Poisson}({\lambda }_1),N_2\sim \mathrm{Poisson}({\lambda }_2),N_1\perp N_2$
• ${\lambda }_1:\text{Rate parameter of }{N}_1$
• ${\lambda }_2:\text{Rate parameter of }{N}_2$
• $u:\text{Uniform[0,1] random varible}$
• $I_{\nu }(x):\text{Modified Bessel function of the first kind of order }\nu$
• $\lfloor x\rfloor :\text{Floor function}$
• $\lceil x\rceil :\text{Ceiling Function}$

Spreadsheet Documents

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