Sebaran Poisson

Dina statistik jeung téori probabilitas, **sebaran Poisson** nyaéta sebaran probabilitas diskrét (dipanggihkeun ku Siméon-Denis Poisson (1781-1840) sarta dipublikasikeun, babarengan jeung téori probabilitas, taun 1838 dina makalahna *Recherches sur la probabilité des jugements en matiéres criminelles et matiére civile*) dumasar kana variabel acak *N* nu diitung, di antara nu séjénna, wilangan kajadian diskrit (kadang kala disebut "datang") nu dicokot salila interval time nu panjang dibéré. Probabilitas nu mana kajadian *k* pasti (*k* salila natural number kaasup 0, *k* = 0, 1, 2, ...) nyaéta:

:P(N=k)=\frac{e^{-\lambda}\lambda^k}{k!}.

nu mana:
* {e^{}} nyaéta dumasar kana logaritma natural  ({e^{}} = 2.71828...),
* {k!} nyaéta faktorial of {k},
* {\lambda} nyaéta wilangan riil positip, sarua jeung wilangan ekspektasi kajadian nu kajadian salila dina interval waktu. Keur contona, lamun kajadian rata-rata unggal minute, sarta museurkeun kana jumlah kajadian unggal 10 menit, mangka bisa ngagunakeunmodél sebaran Poisson ku {\lambda}=5.

## Proses Poisson 
Kadangkala {\lambda} dijadikeun *laju*, dina hal ieu, wilangan rata-rata kajadian per satuan waktu.  Dina kasus eta, lamun *Nt* mangrupa jumlah kajadian saméméh waktu *t* mangka

:P(N_t=k)=\frac{e^{-\lambda t}(\lambda t)^k}{k!},

sarta waktu tunggu *T* ti mimiti kajadian mangrupa variabel random *kontinyu* nu mibanda sebaran eksponensial; probability distribution ieu bisa disimpulkeun tina kanyataan yen

:P(T>t)=P(N_t=0).

Mangsa waktu jadi kalibet, mangka urang mibanda 1-dimensi Poisson process, nu kaasup boh sebaran-Poisson diskrit variabel random nu diitung tina nu datang unggal interval waktu, sarta Erlang-distributed kontinyu waktu tunggu.  Mangka Poisson process dimensi-na leuwih luhur ti 1.

## Kajadian 
Sebaran Poisson diwangun dina pakait jeung proses Poisson. Ilahar dipaké keur rupa-rupa hal nu pakait jeung diskrit alami (saperti hiji kajadian bisa aya dina 0, 1, 2, 3, ... kali salila périodeu waktu atawa daerah nu geus ditangtukeun) iraha waé kamungkinan éta kajadian bakal aya mangrupa hal anu konstan dina waktu atawa ruang. Contona nyaéta :
* Lobana nuclei nu teu stabil, nyaéta "meluruh" dina sawatara waktu nu geus ditangtukeun tina material radioaktif.
* Lobana mobil nu ngaliwat kana hiji titik nu geus ditangtungkeun dina sawatara waktu nu geus ditangtukeun oge.
* Lobana kasalahan ngeja sekretaris waktu ngetik salembar kertas.
* Lobana narima telpon dina sapoe.
* Lilana waktu web server nu noong unggal menit.
** Keur gampangna, lobana editan per jam nu kacatet dina kaca Wikipedia's Parobahan Kiwari nuturkeun distribusi Poisson.
* Lobana roadkill nu kapanggih unggal satuan panjang di jalan.
* Lobana mutation DNA sanggeus kajadian radiasi.
* Lobana tangkal pinus unggal kilométer persegi di leuweung campuran.
* Lobana bentang nu ayana dia volume ruang.
* Lobana tangtara nu palastra ku *horse-kicks* unggal taun dina pasukan berkuda Prussia (contonu kawentar dina buku Ladislaus Josephovich Bortkiewicz (1868-1931)).
* Lobana bom nu ragrag unggal mil pasagi di London salila Jerman nyerang waktu ahir Perang Dunya Kadua.

## How does this distribution arise? -- The limit theorem 
Sebaran binomial mibanda paraméter *n* sarta λ/*n*, dina hal ieu, sebaran probabiliti tina jumlah sukses dina *n* percobaan, mibanda probabiliti λ/*n* tina sukses dina unggal percobaan, ngadeukeutan sebaran Poisson mibanda nilai ekspektasi λ salaku *n* ngadeukeutan tak hingga.

Here are the details.  First, recall from calculus that

:\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda}.

Let *p* = λ/*n*.  Then we have

:\lim_{n\to\infty} P(X=k)=\lim_{n\to\infty}{n \choose x} p^k (1-p)^{n-k}
=\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}

:=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)} \underbrace{\left({\lambda^k \over k!}\right)}\underbrace{\left(1-{\lambda \over n}\right)^n}\underbrace{\left(1-{\lambda \over n}\right)^{-k}}.

As *n* approaches ∞, the expression over the first of the four \underbrace{\mathrm{underbraces}} approaches 1; the expression over the second underbrace remains constant since "*n*" does not appéar in it at all; the expression over the third underbrace approaches *e*−λ; and the one over the fourth underbrace approaches 1.

Consequently the limit is

:{\lambda^k e^{-\lambda} \over k!}.

## Pasipatan 
Nilai ekspektasi variabel random nu kasebar Poisson sarua jeung λ sarta mangrupa varian-na. The higher moments of the Poisson distribution are Touchard polynomials in λ, whose coefficients have a combinatorial méaning.

The most likely value ("mode") of a Poisson distributed random variable is equal to the largest integer ≤ λ, which is also written as floor(λ).

If λ is big enough (λ > 1000 say), then the normal distribution with méan λ and standard deviation √ λ is an excellent approximation to the Poisson distribution. If λ > about 10, then the normal distribution is a good approximation if an appropriate continuity correction is done, i.e., P(*X* ≤ *x*), where (lower-case) *x* is a non-negative integer, is replaced by P(*X* ≤ *x* + 0.5).

If *N* and *M* are two independent random variables, both following a Poisson distribution with paraméters λ and μ, respectively, then *N* + *M* follows a Poisson distribution with paraméter λ + μ.

The moment-generating function of the Poisson distribution with expected value λ is

:E\left(e^{tX}\right)=\sum_{k=0}^\infty e^{tk} P(X=k)=\sum_{k=0}^\infty e^{tk} {\lambda^k e^{-\lambda} \over k!} =e^{\lambda(e^t-1)}.

All of the cumulants of the Poisson distribution are equal to the expected value λ. The *n*th factorial moment of the Poisson distribution is λ*n*.

The Poisson distributions are infinitely divisible probability distributions.

## The "law of small numbers" 
The word **law** is sometimes used as a synonym of probability distribution, and **convergence in law** méans convergence in distribution. Accordingly, the Poisson distribution is sometimes called the **law of small numbers** because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. *The Law of Small Numbers* is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898. Some historians of mathematics have argued that the Poisson distribution should have been called the Bortkiewicz distribution.

## Baca ogé 
* Sebaran Poisson campuran
* Prosés Poisson
* Sebaran Erlang nu ngajelaskeun waktu tunggu salila kajadian n geus kajadian. Keur temporally sebaran kajadian, sebaran Poisson mangrupa sebaran probabiliti wilangan kajadian nu bakal kajadian dina waktu nu ditangtukeun, sebaran Erlang nyaéta sebaran probabiliti antara waktu salila kajadian nu ka-*n*.