User:Noisetau/1

Skellam

Probability mass function Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index k. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters	$\mu _{1}\geq 0,~~\mu _{2}\geq 0$
Support	$\{\ldots ,-2,-1,0,1,2,\ldots \}$
PMF	$e^{-(\mu _{1}\!+\!\mu _{2})}\left({\frac {\mu _{1}}{\mu _{2}}}\right)^{k/2}\!\!I_{k}(2{\sqrt {\mu _{1}\mu _{2}}})$
Mean	$\mu _{1}-\mu _{2}\,$
Median	N/A
Variance	$\mu _{1}+\mu _{2}\,$
Skewness	${\frac {\mu _{1}-\mu _{2}}{(\mu _{1}+\mu _{2})^{3/2}}}$
Excess kurtosis	$1/(\mu _{1}+\mu _{2})\,$
MGF	$e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{t}+\mu _{2}e^{-t}}$
CF	$e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{it}+\mu _{2}e^{-it}}$

Poisson

Probability mass function The horizontal axis is the index k. The function is defined only at integer values of k. The connecting lines are only guides for the eye and do not indicate continuity.
Cumulative distribution function The horizontal axis is the index k.
Parameters	$\lambda \in (0,\infty )$
Support	$k\in \{0,1,2,\ldots \}$
PMF	${\frac {e^{-\lambda }\lambda ^{k}}{k!}}\!$
CDF	${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}\!{\text{ for }}k\geq 0$ (where $\Gamma (x,y)$ is the Incomplete gamma function)
Mean	$\lambda \,$
Median	${\text{usually about }}\lfloor \lambda +1/3-0.02/\lambda \rfloor$
Mode	$\lfloor \lambda \rfloor$ and $\lambda -1$ if $\lambda$ is an integer
Variance	$\lambda \,$
Skewness	$\lambda ^{-1/2}\,$
Excess kurtosis	$\lambda ^{-1}\,$
Entropy	$\lambda [1\!-\!\ln(\lambda )]\!+\!e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\ln(k!)}{k!}}$ (for large $\lambda$) ${\frac {1}{2}}\log(2\pi e\lambda )-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}-{\frac {19}{360\lambda ^{3}}}+O({\frac {1}{\lambda ^{4}}})$
MGF	$\exp(\lambda (e^{t}-1))\,$
CF	$\exp(\lambda (e^{it}-1))\,$