 # 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. 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.

Quick facts: Notation, Parameters, Support, PMF, CDF...
Notation Probability mass function The 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 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. $\operatorname {Pois} (\lambda )$ $\lambda \in (0,\infty )$ (rate) $k\in \mathbb {N} _{0}$ (Natural numbers starting from 0) ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$ ${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloo$ !}},} or $e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }{\frac {\lambda ^{j}}{j!}},$ or $Q(\lfloor k+1\rfloor ,\lambda )$ (for $k\geq 0,$ where $\Gamma (x,y)$ is the upper incomplete gamma function, $\lfloor k\rfloor$ is the floor function, and $Q$ is the regularized gamma function) $\lambda$ $\approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor$ $\left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor$ $\lambda$ ${\frac {1}{\sqrt {\lambda }}}$ ${\frac {1}{\lambda }}$ $\lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}$ or for large $\lambda$ {\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}} $\exp \left[\lambda \left(e^{t}-1\right)\right]$ $\exp \left[\lambda \left(e^{it}-1\right)\right]$ $\exp \left[\lambda \left(z-1\right)\right]$ ${\frac {1}{\lambda }}$ Close

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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