= Poisson distribution
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{title2=$\operatorname{Pois}(\lambda)$}
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= Poisson random variable
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= Poisson distributed
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= Poisson
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For $\lambda>0$, a <Poisson random variable> $N$ has <probability mass function>
$$
\mathbb P(N=k)=e^{-\lambda}\frac{\lambda^k}{k!},\quad k=0,1,2,\ldots,
\qquad \mathbb E N=\operatorname{Var}(N)=\lambda.
$$
The <exponential function> series verifies normalization. At $\lambda=0$, define $N=0$ with certainty. The equality of <mean> and <variance> is a distinguishing model property, though it does not by itself characterize the law.
When counting events in a homogeneous <Poisson process>, distinguish its rate $r$ from the distribution parameter: an interval of duration $t$ has $\lambda=rt$. A rate measured per minute times a duration measured in minutes gives a dimensionless expected count. This process assumes <independent increments>, a constant rate, and a negligible chance of multiple events relative to interval length in a sufficiently short interval. A rate alone does not establish a <Poisson distribution>. See https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/d973b10c2587781f86ca4f2aff49098f_MITRES_6_012S18_Textbook.pdf[Bertsekas and Tsitsiklis, chapter 6 summary, pages 46-47].
For a constructed arrival model with $r=2$ events per minute over $t=1.5$ minutes, $\lambda=3$. The <mean> and <variance> are both three, and
$$
\mathbb P(N=0)=e^{-3}\approx0.0497871,\qquad
\mathbb P(N=3)=e^{-3}\frac{3^3}{3!}=4.5e^{-3}\approx0.2240418.
$$
The <probability> of at least one event is $1-e^{-3}\approx0.9502129$. These results describe the specified process, not every arrival mechanism with the same average rate.
The <Poisson limit theorem>, often called the law of rare events, connects this law to the <binomial distribution>. Keep $\lambda>0$ fixed and let $K_n\sim\operatorname{Bin}(n,\lambda/n)$ for integers $n\geq\lambda$. For each fixed nonnegative integer $k$ and sufficiently large $n\geq k$,
$$
\mathbb P(K_n=k)=\frac{\lambda^k}{k!}
\left(\prod_{j=0}^{k-1}\left(1-\frac jn\right)\right)
\left(1-\frac\lambda n\right)^{n-k}
\longrightarrow e^{-\lambda}\frac{\lambda^k}{k!}.
$$
There are more opportunities, each less likely to succeed, while the <mean> stays $\lambda$ and the <variance> $\lambda(1-\lambda/n)$ approaches $\lambda$. For $\lambda=3$, the <probability> of three successes is $0.2668279$ at $n=10$, $0.2310569$ at $n=50$, and $0.2257398$ at $n=200$, approaching the <Poisson distribution> value $0.2240418$.
\Image[/binomial-poisson-limit.png]
{title=Binomial counts approach Poisson counts while their mean stays fixed at three}
The panels show the same count scale and the same <Poisson distribution> reference. At $n=10$, $p=0.3$ is not especially small and the difference is visible. No universal trial-count cutoff guarantees an adequate approximation for every event.
The <independence> assumption matters even when every success is rare. For $n\geq3$, if all $n$ <indicator random variables> equal one shared <Bernoulli random variable> of parameter $3/n$, their sum is zero with <probability> $1-3/n$ and $n$ with <probability> $3/n$. Its <mean> is three, yet its zero-count <probability> tends to one, not $e^{-3}$. Some dependent counts still admit approximation through the <Stein-Chen method>, but this requires additional control of dependence.
For historical orientation, the BnF records Poisson's original probability treatise, \i[Recherches sur la probabilité des jugements], as published in Paris in 1837. This bibliographic fact alone does not establish when the modern distribution terminology arose. See https://catalogue.bnf.fr/ark:/12148/cb31129269s[BnF record of Poisson's 1837 treatise].
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