# Poisson distribution

## Discrete probability distribution / From Wikipedia, the free encyclopedia

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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.

**Quick facts: Notation, Parameters, Support, PMF, CDF...**▼

Probability mass function | |||

Cumulative distribution function | |||

Notation | $\operatorname {Pois} (\lambda )$ | ||
---|---|---|---|

Parameters | $\lambda \in (0,\infty )$ (rate) | ||

Support | $k\in \mathbb {N} _{0}$ (Natural numbers starting from 0) | ||

PMF | ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$ | ||

CDF |
${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor$ !}},} 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) | ||

Mean | $\lambda$ | ||

Median | $\approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor$ | ||

Mode | $\left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor$ | ||

Variance | $\lambda$ | ||

Skewness | ${\frac {1}{\sqrt {\lambda }}}$ | ||

Ex. kurtosis | ${\frac {1}{\lambda }}$ | ||

Entropy |
$\lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}$ or for large $\lambda$ ${\begin{aligned}\approx {\frac {1}{2}}\log \left(2\pi e\lambda \right)-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}\\-{\frac {19}{360\lambda ^{3}}}+{\mathcal {O}}\left({\frac {1}{\lambda ^{4}}}\right)\end{aligned}}$ | ||

MGF | $\exp \left[\lambda \left(e^{t}-1\right)\right]$ | ||

CF | $\exp \left[\lambda \left(e^{it}-1\right)\right]$ | ||

PGF | $\exp \left[\lambda \left(z-1\right)\right]$ | ||

Fisher information | ${\frac {1}{\lambda }}$ |

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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