# Poisson distreebution

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Parameters Probability mass function The horizontal axis is the index k, the nummer o occurrences. λ is the expectit value. The function is defined anly at integer values o k. The connectin lines are anly guides for the ee. Cumulative distribution function The horizontal axis is the index k, the number o occurrences. The CDF is discontinuous at the integers o k an flat everywhaur ense acause a variable that is Poisson distreebutit taks on anly integer values. λ > 0 (real) k ∈ ℤ* ${\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}}$ ${\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}}$, or ${\displaystyle e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\ }$, or ${\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}$ (for ${\displaystyle k\geq 0}$, whaur ${\displaystyle \Gamma (x,y)}$ is the incomplete gamma function, ${\displaystyle \lfloor k\rfloor }$ is the fluir function, an Q is the regularised gamma function) ${\displaystyle \lambda }$ ${\displaystyle \approx \lfloor \lambda +1/3-0.02/\lambda \rfloor }$ ${\displaystyle \lceil \lambda \rceil -1,\lfloor \lambda \rfloor }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda ^{-1/2}}$ ${\displaystyle \lambda ^{-1}}$ ${\displaystyle \lambda [1-\log(\lambda )]+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}$ (for large ${\displaystyle \lambda }$) ${\displaystyle {\frac {1}{2}}\log(2\pi e\lambda )-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}-}$ ${\displaystyle \qquad {\frac {19}{360\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)}$ ${\displaystyle \exp(\lambda (e^{t}-1))}$ ${\displaystyle \exp(\lambda (e^{it}-1))}$ ${\displaystyle \exp(\lambda (z-1))}$ ${\displaystyle \lambda ^{-1}}$

In probability theory an stateestics, the Poisson distreebution (French pronunciation [pwasɔ̃]), named efter French mathematician Siméon Denis Poisson, is a discrete probability distreebution that expresses the probability o a gien nummer o events occurrin in a fixed interval o time an/or space if thir events occur wi a kent average rate an independently o the time syne the last event.[1]

## References

1. Frank A. Haight (1967). Handbook of the Poisson Distribution. New York: John Wiley & Sons.