Poisson Distribution (Pois₍λ₎) ○◂｜appendage｜___｜20260628174208-01-•

Poisson distribution	…	…	…
Probability mass functionThe horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye.	…	…	…
Cumulative distribution functionThe horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values.	…	…	…
Notation	$\mathrm{Pois}(\lambda )$	…	…
Parameters	$\lambda \in (0,\infty )$ (rate)	…	…
Support	$k\in {\mathbb{N}}_0$ (Natural numbers starting from 0)	…	…
PMF	$\frac{{\lambda }^k{e}^{-\lambda }}{k!}$	…	…
CDF	$\frac{\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !},$ or $e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }\frac{{\lambda }^j}{j!},$ or $Q(\lfloor k+1\rfloor ,\lambda )$ (for $k\ge 0,$ where $\Gamma (x,y)$ is the upper incomplete gamma function, $\lfloor k\rfloor$ is the floor function, and $Q$ is the regularized gamma function)	…	…
Mean	$\lambda$	…	…
Median	$\approx \lfloor \lambda +\frac{1}{3}-\frac{1}{50\lambda }\rfloor$	…	…
Mode	$\lceil \lambda \rceil -1,\lfloor \lambda \rfloor$	…	…
Variance	$\lambda$	…	…
Skewness	$\frac{1}{\sqrt{\lambda }}$	…	…
Excess kurtosis	$\frac{1}{\lambda }$	…	…
Entropy	$\lambda [1-\ln (\lambda )]+e^{-\lambda }\sum _{k=0}^{\infty }\frac{{\lambda }^k\ln (k!)}{k!}$ or for large $\lambda$ $\begin{array}{r}\approx \frac{1}{2}\ln \left(2\pi e\lambda \right)-\frac{1}{12\lambda }-\frac{1}{24{\lambda }^2}\\ -\frac{19}{360{\lambda }^3}+\mathcal{O}\left(\frac{1}{{\lambda }^4}\right)\end{array}$	…	…
MGF	$\exp \left[\lambda \left(e^t-1\right)\right]$	…	…
CF	$\exp \left[\lambda \left(e^{it}-1\right)\right]$	…	…
PGF	$\exp \left[\lambda \left(z-1\right)\right]$	…	…
Fisher information	$\frac{1}{\lambda }$	…	…

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