Sebaran Poisson

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Dina statistik jeung téori probabilitas, sebaran Poisson nyaeta sebaran probabilitas diskrét (dipanggihkeun ku Siméon-Denis Poisson (1781-1840) sarta dipublikasikeun, babarengan jeung teori probabilitas, taun 1838 dina makalahna Recherches sur la probabilité des jugements en matières criminelles et matière civile) dumasar kana variabel acak N nu diitung, diantara nu sejenna, wilangan kajadian diskrit (kadang kala disebut "datang") nu dicokot salila interval time nu panjang dibere. Probabilitas numana kajadian k pasti (k salila natural number kaasup 0, k = 0, 1, 2, ...) nyaeta:

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

Numana:

$e$ nyaeta dumasar kana logaritma natural ( $e$ = 2.71828...),
$k!$ nyaeta faktorial of $k$ ,
$λ$ nyaeta 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 migunakeun model sebaran Poisson ku $λ = 5$ .

Daptar eusi

1 Proses Poisson
2 Kajadian
3 How does this distribution arise? -- The limit theorem
4 Pasipatan
5 The "law of small numbers"
6 Baca ogé

[édit] Proses Poisson

Kadangkala $λ$ dijadikeun laju, dina hal ieu, wilangan rata-rata kajadian per satuan waktu. Dina kasus eta, lamun N_t ngarupakeun jumlah kajadian samemeh waktu t mangka

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

sarta waktu tunggu T ti mimiti kajadian ngarupakeun 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.

[édit] Kajadian

Sebaran Poisson diwangun dina pakait jeung proses Poisson. Ilahar dipake keur rupa-rupa hal nu pakait jeung diskrit alami (saperti hiji kajadian bisa aya dina 0, 1, 2, 3, ... kali salila periode waktu atawa daerah nu geus ditangtukeun) iraha wae kamungkinan eta kajadian bakal aya ngarupakeun hal anu konstan dina waktu atawa ruang. Contona nyaeta :

The number of unstable nuclei that decayed within a given period of time in a piece of radioactive substance.
The number of cars that pass through a certain point on a road during a given period of time.
The number of spelling mistakes a secretary makes while typing a single page.
The number of phone calls you get per day.
The number of times your web server is accessed per minute.
- For instance, the number of edits per hour recorded on Wikipedia's Recent Changes page follows an approximately Poisson distribution.
The number of roadkill you find per unit length of road.
The number of mutations in a given stretch of DNA after a certain amount of radiation.
The number of pine trees per square mile of mixed forest.
The number of stars in a given volume of space.
The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry (an example made famous by a book of Ladislaus Josephovich Bortkiewicz (1868-1931)).
The number of bombs falling on each square mile of London during a German air raid in the early part of the Second World War.

[édit] How does this distribution arise? -- The limit theorem

Sebaran binomial mibanda parameter 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 appear 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

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

[édit] Pasipatan

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

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 mean 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 parameters λ and μ, respectively, then N + M follows a Poisson distribution with parameter 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 nth factorial moment of the Poisson distribution is λⁿ.

The Poisson distributions are infinitely divisible probability distributions.

[édit] The "law of small numbers"

The word law is sometimes used as a synonym of probability distribution, and convergence in law means 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.

[édit] Baca ogé

Sebaran Poisson campuran
Prosés Poisson
Sebaran Erlang nu ngajelaskeun waktu tunggu salila kajadian n geus kajadian. Keur temporally sebaran kajadian, sebaran Poisson ngarupakeun sebaran probabiliti wilangan kajadian nu bakal kajadian dina waktu nu ditangtukeun, sebaran Erlang nyaeta sebaran probabiliti antara waktu salila kajadian nu ka-n.

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