# Poisson distreebution

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* ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$ ${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}$ , or $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ , or $Q(\lfloor k+1\rfloor ,\lambda )$ (for $k\geq 0$ , whaur $\Gamma (x,y)$ is the incomplete gamma function, $\lfloor k\rfloor$ is the fluir function, an Q is the regularised gamma function) $\lambda$ $\approx \lfloor \lambda +1/3-0.02/\lambda \rfloor$ $\lceil \lambda \rceil -1,\lfloor \lambda \rfloor$ $\lambda$ $\lambda ^{-1/2}$ $\lambda ^{-1}$ $\lambda [1-\log(\lambda )]+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}$ (for large $\lambda$ ) ${\frac {1}{2}}\log(2\pi e\lambda )-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}-$ $\qquad {\frac {19}{360\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)$ $\exp(\lambda (e^{t}-1))$ $\exp(\lambda (e^{it}-1))$ $\exp(\lambda (z-1))$ $\lambda ^{-1}$ 