# Poisson distribution

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In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.[1] It is named after French mathematician Siméon Denis Poisson (/ˈpwɑːsɒn/; French pronunciation: [pwasɔ̃]). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. It plays an important role for discrete-stable distributions.

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Notation 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. ${\displaystyle \operatorname {Pois} (\lambda )}$ ${\displaystyle \lambda \in (0,\infty )}$ (rate) ${\displaystyle k\in \mathbb {N} _{0}}$ (Natural numbers starting from 0) ${\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}}$ ${\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor$ !}},} or ${\displaystyle e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }{\frac {\lambda ^{j}}{j!}},}$ or ${\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}$ (for ${\displaystyle k\geq 0,}$ where ${\displaystyle \Gamma (x,y)}$ is the upper incomplete gamma function, ${\displaystyle \lfloor k\rfloor }$ is the floor function, and ${\displaystyle Q}$ is the regularized gamma function) ${\displaystyle \lambda }$ ${\displaystyle \approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor }$ ${\displaystyle \left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor }$ ${\displaystyle \lambda }$ ${\displaystyle {\frac {1}{\sqrt {\lambda }}}}$ ${\displaystyle {\frac {1}{\lambda }}}$ ${\displaystyle \lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}$   or for large ${\displaystyle \lambda }$   {\displaystyle {\begin{aligned}\approx {\frac {1}{2}}\log \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{aligned}}} ${\displaystyle \exp \left[\lambda \left(e^{t}-1\right)\right]}$ ${\displaystyle \exp \left[\lambda \left(e^{it}-1\right)\right]}$ ${\displaystyle \exp \left[\lambda \left(z-1\right)\right]}$ ${\displaystyle {\frac {1}{\lambda }}}$
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For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3. The most likely number of calls received are 2 and 3, but 1 and 4 are also likely. There is a small probability of it being as low as zero and a very small probability it could be 10 or even higher.

Another example is the number of decay events that occur from a radioactive source during a defined observation period.

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