Poisson distribution

Poisson
Probability mass function The horizontal axis is the index k, the number of occurrences. λ is the expected number 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 function The 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.
Parameters	λ > 0 (real)
	k ∈ ℕ
pmf	${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$ ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$
CDF	${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}$ ${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}$ , or $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ , or $Q(\lfloor k+1\rfloor ,\lambda )$ $Q(\lfloor k+1\rfloor ,\lambda )$ (for $k\geq 0$ $k\geq 0$ , where $\Gamma (x,y)$ $\Gamma (x,y)$ is the incomplete gamma function, $\lfloor k\rfloor$ $\lfloor k\rfloor$ is the floor function, and Q is the regularized gamma function)
Mean	$\lambda$ $\lambda$
Median	$\approx \lfloor \lambda +1/3-0.02/\lambda \rfloor$ $\approx \lfloor \lambda +1/3-0.02/\lambda \rfloor$
	$\lceil \lambda \rceil -1,\lfloor \lambda \rfloor$ $\lceil \lambda \rceil -1,\lfloor \lambda \rfloor$
Variance	$\lambda$ $\lambda$
Skewness	$\lambda ^{-1/2}$ $\lambda ^{-1/2}$
Ex. kurtosis	$\lambda ^{-1}$ $\lambda ^{-1}$
Entropy	$\lambda [1-\log(\lambda )]+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}$ $\lambda [1-\log(\lambda )]+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}$ (for large $\lambda$ $\lambda$ ) ${\frac {1}{2}}\log(2\pi e\lambda )-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}-$ ${\frac {1}{2}}\log(2\pi e\lambda )-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}-$ $\qquad {\frac {19}{360\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)$ $\qquad {\frac {19}{360\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)$
MGF	$\exp(\lambda (e^{t}-1))$ $\exp(\lambda (e^{t}-1))$
	$\exp(\lambda (e^{it}-1))$ $\exp(\lambda (e^{it}-1))$
PGF	$\exp(\lambda (z-1))$ $\exp(\lambda (z-1))$
Fisher information	${\frac {1}{\lambda }}$ $\frac{1}{\lambda}$

${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}$ ${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}$ , or $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ , or $Q(\lfloor k+1\rfloor ,\lambda )$ $Q(\lfloor k+1\rfloor ,\lambda )$

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