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.)
Cumulative distribution function
Parameters	$\mu_1\ge 0,~~\mu_2\ge 0$
Support	$\{\ldots, -2,-1,0,1,2,\ldots\}$
Probability mass function (pmf)	$e^{-(\mu_1\!+\!\mu_2)} \left(\frac{\mu_1}{\mu_2}\right)^{k/2}\!\!I_k(2\sqrt{\mu_1\mu_2})$
Cumulative distribution function (cdf)
Mean	$\mu_1-\mu_2\,$
Median	N/A
Mode
Variance	$\mu_1+\mu_2\,$
Skewness	$\frac{\mu_1-\mu_2}{(\mu_1+\mu_2)^{3/2}}$
Excess kurtosis	$1/(\mu_1+\mu_2)\,$
Entropy
Moment-generating function (mgf)	$e^{-(\mu_1+\mu_2)+\mu_1e^t+\mu_2e^{-t}}$
Characteristic function	$e^{-(\mu_1+\mu_2)+\mu_1e^{it}+\mu_2e^{-it}}$

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	$k \in \{0,1,2,\ldots\}$
Probability mass function (pmf)	$\frac{e^{-\lambda} \lambda^k}{k!}\!$
Cumulative distribution function (cdf)	$\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\!\text{ for }k\ge 0$ (where $Γ(x, y)$ is the Incomplete gamma function)
Mean	$\lambda\,$
Median	$\text{usually about }\lfloor\lambda+1/3-0.02/\lambda\rfloor$
Mode	$\lfloor\lambda\rfloor$ and $λ - 1$ if $λ$ 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 $λ$ ) $\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} - \frac{19}{360 \lambda^3} + O(\frac{1}{\lambda^4})$
Moment-generating function (mgf)	$\exp(\lambda (e^t-1))\,$
Characteristic function	$\exp(\lambda (e^{it}-1))\,$