Poisson distribution

Poisson
Probability mass function The horizontal axis is the index k, the number of occurrences. The function is only defined 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 only takes on integer values.
Notation	$\mathrm{Pois}(\lambda)\,$
Parameters	λ > 0 (real)
Support	k ∈ { 0, 1, 2, 3, ... }
pmf	$\frac{\lambda^k}{k!}\cdot e^{-\lambda}$
CDF	$\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\!$ --or-- $e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\$ (for $k\ge 0$ where $\Gamma(x, y)\,\!$ is the Incomplete gamma function and $\lfloor k\rfloor$ is the floor function)
Mean	$\lambda\,\!$
Median	$\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor$
Mode	$\lfloor\lambda\rfloor,\,\lceil\lambda\rceil - 1$
Variance	$\lambda\,\!$
Skewness	$\lambda^{-1/2}\,$
Ex. kurtosis	$\lambda^{-1}\,$
Entropy	$\lambda[1\!-\!\log(\lambda)]\!+\!e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}$ (for large $\lambda$ ) $\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -$ $\frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)$
MGF	$\exp(\lambda (e^{t}-1))\,$
CF	$\exp(\lambda (e^{it}-1))\,$
PGF	$\exp(\lambda(z - 1))\,$

In probability theory and statistics, the Poisson distribution (pronounced [pwasɔ̃]) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.^[1] The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

For instance, suppose someone typically gets 4 pieces of mail per day on average. There will be, however, a certain spread: sometimes a little more, sometimes a little less, once in a while nothing at all.^[2] Given only the average rate, for a certain period of observation (pieces of mail per day, phonecalls per hour, etc.), and assuming that the process, or mix of processes, that produces the event flow is essentially random, the Poisson distribution specifies how likely it is that the count will be 3, or 5, or 10, or any other number, during one period of observation. That is, it predicts the degree of spread around a known average rate of occurrence.^[2]

The Derivation of the Poisson distribution section shows the relation with a formal definition.

Historical background of the Poisson distribution was described by Gullberg (1997).^[3]

History[edit]

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published, together with his probability theory, in 1837 in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (“Research on the Probability of Judgments in Criminal and Civil Matters”).^[4] The work focused on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or “arrivals”) that take place during a time-interval of given length. The result had been given previously by Abraham de Moivre (1711) in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus in Philosophical Transactions of the Royal Society, p. 219.^[5]

A practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kick; this experiment introduced the Poisson distribution to the field of reliability engineering.^[6]

Definition[edit]

A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if for k = 0, 1, 2, ... the probability mass function of X is given by:

$\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k e^{-\lambda}}{k!},$

where

e is the base of the natural logarithm (e = 2.71828...)
k! is the factorial of k.
$\lambda=\lambda T,$ when the number of events occurring will be observed in the time interval $T=1.$ ^[7]

The positive real number λ is equal to the expected value of X and also to its variance^[8]

$\lambda=\operatorname{E}(X)=\operatorname{Var}(X).$

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The Poisson distribution is sometimes called a Poissonian.

Properties[edit]

Mean[edit]

The expected value of a Poisson-distributed random variable is equal to λ and so is its variance.
The coefficient of variation is $\textstyle \lambda^{-1/2}$ , while the index of dispersion is 1.^[5]
The mean deviation about the mean is^[5]

$\operatorname{E}|X-\lambda|= 2\exp(-\lambda) \frac{\lambda^{\lfloor\lambda\rfloor + 1}}{ \lfloor\lambda\rfloor!} .$

The mode of a Poisson-distributed random variable with non-integer λ is equal to $\scriptstyle\lfloor \lambda \rfloor$ , which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ is a positive integer, the modes are λ and λ − 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.

Median[edit]

Bounds for the median (ν) of the distribution are known and are sharp:^[9]

$\lambda - \ln{2} \le \nu < \lambda + \frac{1}{3}.$

Higher moments[edit]

The higher moments m_k of the Poisson distribution about the origin are Touchard polynomials in λ:

$m_k = \sum_{i=1}^k \lambda^i \left\{\begin{matrix} k \\ i \end{matrix}\right\},$

where the {braces} denote Stirling numbers of the second kind.^[10] The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth moment equals the number of partitions of a set of size n.

Sums of Poisson-distributed random variables:

If $X_i \sim \mathrm{Pois}(\lambda_i)\, i=1,\dots,n$ are independent, and $\lambda=\sum_{i=1}^n \lambda_i$ , then $Y = \left( \sum_{i=1}^n X_i \right) \sim \mathrm{Pois}(\lambda)$ .^[11] A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so is each of those two independent random variables.^[12]

Other properties[edit]

The Poisson distributions are infinitely divisible probability distributions.^[13]^[14]
The directed Kullback–Leibler divergence of Pois(λ₀) from Pois(λ) is given by

$D_{\mathrm{KL}}(\lambda\|\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.$

Bounds for the tail probabilities of a Poisson random variable $X \sim \text{Pois}(\lambda)$ can be derived using a Chernoff bound argument.^[15]

$P(X \geq x) \leq \frac{e^{-\lambda} (e \lambda)^x}{x^x}, \text{ for } x > \lambda ,$

$P(X \leq x) \leq \frac{e^{-\lambda} (e \lambda)^x}{x^x}, \text{ for } x < \lambda .$

Related distributions[edit]

If $X_1 \sim \mathrm{Pois}(\lambda_1)\,$ and $X_2 \sim \mathrm{Pois}(\lambda_2)\,$ are independent, then the difference $Y = X_1 - X_2$ follows a Skellam distribution.
If $X_1 \sim \mathrm{Pois}(\lambda_1)\,$ and $X_2 \sim \mathrm{Pois}(\lambda_2)\,$ are independent, then the distribution of $X_1$ conditional on $X_1+X_2$ is a binomial distribution. Specifically, given $X_1+X_2=k$ , $\!X_1\sim \mathrm{Binom}(k, \lambda_1/(\lambda_1+\lambda_2))$ . More generally, if X₁, X₂,..., X_n are independent Poisson random variables with parameters λ₁, λ₂,..., λ_n then

given $\sum_{j=1}^n X_j=k,$ $X_i \sim \mathrm{Binom}\left(k, \frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right)$ . In fact, $\{X_i\} \sim \mathrm{Multinom}\left(k, \left\{\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right\}\right)$ .

If $X \sim \mathrm{Pois}(\lambda)\,$ and the distribution of $Y$ , conditional on X = k, is a binomial distribution, $Y | (X = k) \sim \mathrm{Binom}(k, p)$ , then the distribution of Y follows a Poisson distribution $Y \sim \mathrm{Pois}(\lambda \cdot p)\,$ . In fact, if $\{Y_i\}$ , conditional on X = k, follows a multinomial distribution, $\{Y_i\} | (X = k) \sim \mathrm{Multinom}\left(k, p_i\right)$ , then each $Y_i$ follows an independent Poisson distribution $Y_i \sim \mathrm{Pois}(\lambda \cdot p_i), \rho(Y_i, Y_j) = 0$ .
The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.^[16]

$F_\mathrm{Binomial}(k;n, p) \approx F_\mathrm{Poisson}(k;\lambda=np)\,$

The Poisson distribution is a special case of generalized stuttering Poisson distribution (or stuttering Poisson distribution) with only a parameter.^[17] The stuttering Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a special case of a compound Poisson distribution.
For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ and variance λ (standard deviation $\sqrt{\lambda}$ ), is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., P(X ≤ x), where (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5).

$F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$

Variance-stabilizing transformation: When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about $\sqrt \lambda$ and variance of about 1/4.^[18]^[19] Under this transformation, the convergence to normality (as λ increases) is far faster than the untransformed variable.^{[citation needed]} Other, slightly more complicated, variance stabilizing transformations are available,^[19] one of which is Anscombe transform. See Data transformation (statistics) for more general uses of transformations.
If for every t > 0 the number of arrivals in the time interval [0,t] follows the Poisson distribution with mean λ t, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1 / λ.^[20]
The cumulative distribution functions of the Poisson and chi-squared distributions are related in the following ways:^[21]

$F_\text{Poisson}(k;\lambda) = F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,$

and^[22]

$\Pr(X=k)=F_{\chi^2}(2\lambda;2k) -F_{\chi^2}(2\lambda;2(k+1)) .$

Occurrence[edit]

Applications of the Poisson distribution can be found in many fields related to counting:^[23]

Telecommunication example: telephone calls arriving in a system.
Astronomy example: photons arriving at a telescope.
Biology example: the number of mutations on a strand of DNA per unit length.
Management example: customers arriving at a counter or call centre.
Civil engineering example: cars arriving at a traffic light.
Finance and insurance example: number of Losses/Claims occurring in a given period of Time.
Earthquake seismology example: an asymptotic Poisson model of seismic risk for large earthquakes. (Lomnitz, 1994).
Radioactivity Example: Decay of a radioactive nucleus.

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:

The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
The number of yeast cells used when brewing Guinness beer. This example was made famous by William Sealy Gosset (1876–1937).^[24]
The number of phone calls arriving at a call centre per minute.
The number of goals in sports involving two competing teams.
The number of deaths per year in a given age group.
The number of jumps in a stock price in a given time interval.
Under an assumption of homogeneity, the number of times a web server is accessed per minute.
The number of mutations in a given stretch of DNA after a certain amount of radiation.
The proportion of cells that will be infected at a given multiplicity of infection.
The arrival of photons on a pixel circuit at a given illumination and over a given time period.
The targeting of V-1 flying bombs on London during World War II.^[25]

Derivation of Poisson distribution — The law of rare events[edit]

Multi-dimensional Poisson process[edit]

Main article: Poisson process

The poisson distribution arises as the distribution of counts of occurrences of events in (multidimensional) intervals in multidimensional Poisson processes in a directly equivalent way to the result for unidimensional processes. This,is D is any region the multidimensional space for which |D|, the area or volume of the region, is finite, and if N(D) is count of the number of events in D, then

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

Other applications in science[edit]

In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a standard deviation $\sigma_k =\sqrt{\lambda}$ . These fluctuations are denoted as Poisson noise or (particularly in electronics) as shot noise.^{[citation needed]}

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If N electrons pass a point in a given time t on the average, the mean current is $I=eN/t$ ; since the current fluctuations should be of the order $\sigma_I=e\sqrt{N}/t$ (i.e., the standard deviation of the Poisson process), the charge $e$ can be estimated from the ratio $t\sigma_I^2/I$ .^{[citation needed]}

An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).^{[citation needed]} Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of receptor molecules in a cell membrane.

$\Pr(N_t=k) = f(k;\lambda t) = \frac{e^{-\lambda t} (\lambda t)^k}{k!} .$

In Causal Set theory the discrete elements of spacetime follow a Poisson distribution in the volume.

Generating Poisson-distributed random variables[edit]

A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth (see References below):

algorithm poisson random number (Knuth):
    init:
         Let L ← e^−λ, k ← 0 and p ← 1.
    do:
         k ← k + 1.
         Generate uniform random number u in [0,1] and let p ← p × u.
    while p > L.
    return k − 1.

While simple, the complexity is linear in λ. There are many other algorithms to overcome this. Some are given in Ahrens & Dieter, see References below. Also, for large values of λ, there may be numerical stability issues because of the term e^−λ. One solution for large values of λ is rejection sampling, another is to use a Gaussian approximation to the Poisson.

Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.

Parameter estimation[edit]

Maximum likelihood[edit]

Given a sample of n measured values k_i we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is ^[28]

$\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i. \!$

Since each observation has expectation λ so does this sample mean. Therefore the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator, i.e. its estimation variance achieves the Cramér–Rao lower bound (CRLB).^{[citation needed]} Hence it is MVUE. Also it can be proved that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one which depends solely on the sample $\mathbf{x}$ (called $h(\mathbf{x})$ ) and one which depends on the parameter $\lambda$ and the sample $\mathbf{x}$ only through the function $T(\mathbf{x})$ . Then, $T(\mathbf{x})$ is a sufficient statistic for $\lambda$ .

$P(\mathbf{x})=\prod_{i=1}^n\frac{\lambda^x e^{-\lambda}}{x!}=\frac{1}{\prod_{i=1}^n x_i!} \times \lambda^{\sum_{i=1}^n x_i}e^{-n\lambda}$

Note that the first term, $h(\mathbf{x})$ , depends only on $\mathbf{x}$ . The second term, $g(T(\mathbf{x})|\lambda)$ , depends on the sample only through $T(\mathbf{x})=\sum_{i=1}^nx_i$ . Thus, $T(\mathbf{x})$ is sufficient.

For completeness, a family of distributions is said to be complete if and only if $E(g(T)) = 0$ implies that $P_\lambda(g(T)) = 0) = 1$ for all $\lambda$ . If the individual $X_i$ are iid $Po(\lambda)$ , then $T(\mathbf{x})=\sum_{i=1}^nX_i\sim Po(n\lambda)$ . Knowing the distribution we want to investigate it is easy to see that the statistic is complete.

$E(g(T))=\sum_{t=0}^\infty g(t)\frac{(n\lambda)^te^{-n\lambda}}{t!}=0$

For this equality to hold, it is obvious that $g(t)$ must be 0. This follows from the fact that none of the other terms will be 0 for all $t$ in the sum and for all possible values of $\lambda$ . Hence, $E(g(T)) = 0$ for all $\lambda$ implies that $P_\lambda(g(T)) = 0) = 1$ and the statistic has been shown to be complete.

Confidence interval[edit]

The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level $1 - α$ is

$\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2),$

or equivalently,

$F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),$

where $\chi^{2}(p;n)$ is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and $F^{-1}(p;n,1)$ is the quantile function of a Gamma distribution with shape parameter n and scale parameter 1.^[21]^[29] This interval is 'exact' in the sense that its coverage probability is never less than the nominal $1 - α$ .

When quantiles of the Gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):^[30]

$k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3,$

where $z_{\alpha/2}$ denotes the standard normal deviate with upper tail area $α / 2$ .

For application of these formulae in the same context as above (given a sample of n measured values k_i each drawn from a Poisson distribution with mean λ), one would set

$k=\sum_{i=1}^n k_i ,\!$

calculate an interval for μ=nλ, and then derive the interval for λ.

Bayesian inference[edit]

In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution.^[31] Let

$\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!$

denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:

$g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.$

Then, given the same sample of n measured values k_i as before, and a prior of Gamma(α, β), the posterior distribution is

$\lambda \sim \mathrm{Gamma}(\alpha + \sum_{i=1}^n k_i, \beta + n). \!$

The posterior mean E[λ] approaches the maximum likelihood estimate $\widehat{\lambda}_\mathrm{MLE}$ in the limit as $\alpha\to 0,\ \beta\to 0$ .^{[citation needed]}

The posterior predictive distribution for a single additional observation is a negative binomial distribution,^{[citation needed]} sometimes called a Gamma–Poisson distribution.

Bivariate Poisson distribution[edit]

This distribution has been extended to the bivariate case.^[32] The generating function for this distribution is

$g( u, v ) = \exp[ ( \theta_1 - \theta_{ 12 } )( u - 1 ) + ( \theta_2 - \theta_{ 12 } )( v - 1 ) + \theta_{ 12 } ( uv - 1 ) ]$

with

$\theta_1, \theta_2 > \theta_{ 12 } > 0 \,$

The marginal distributions are Poisson(θ₁) and Poisson(θ₂) and the correlation coefficient is limited to the range

$0 \le \rho \le \min\left\{ \frac{ \theta_1 }{ \theta_2 }, \frac{ \theta_2 }{ \theta_1 } \right\}$

The Skellam distribution is a particular case of this distribution.^{[citation needed]}

Poisson distribution and prime numbers[edit]

Gallagher in 1976 showed that prime numbers in short intervals obey a Poisson distribution.^[33] Ernie Croot(2010) stated this in informal mathematical language in his lecture notes on the Poisson distribution.^[34] To understand this relationship the Prime Number Theorem will be required.

This theorem states that the number of primes $p \le x$ is about $x/\log x$ , where the logarithm is take to the base e. In symbols if $\pi (x)$ denotes the number of primes less than x then

$\pi (x) \sim \frac{x}{\log x},$

This implies that

$\lim_{ x \to \infty} \frac{ \pi ( x ) } { x/ \log x } = 1.$

In what follows the notation $p \le x$ is used to denote the number of primes in a given interval $I$ .

Suppose that $x$ is a large number, say $10^{ 100 }$ . Then a number chosen at random $n \le x$ has $\sim 1/100 \log(10)$ (~ 0.43%) chance that it will be prime. A typical interval $[ n,\ n + \log n ] \subset [ 1,\ x ]$ will contain about one prime.

More than this is true: If a number $n \le x$ is chosen at random, and choose $\lambda \ge 0$ and $j$ not too large ( say $\lambda ,j \le 20$ ) then the number of primes in $[ n , n + \log(n) ]$ is approximately Poisson-distributed:

$P\{ \pi [ n,{ \rm{ } } n + \lambda \log n ] = j\} \approx \frac{ e^{-\lambda} \lambda ^j }{j!}$

Notice that equality was not used here: in order to obtain equality we would have to let $x \to \infty$ in some fashion. The larger $x$ is the closer the above probability comes to $\frac{ e^{-\lambda} \lambda^j }{ j! }$ .

It is an interesting exercise to determine out why the primes would be expected to be Poisson-distributed.

Notes[edit]

^ Frank A. Haight (1967). Handbook of the Poisson Distribution. New York: John Wiley & Sons.
^ ^a ^b "Statistics | The Poisson Distribution". Umass.edu. 2007-08-24. Retrieved 2012-04-05.
^ Gullberg, Jan (1997). Mathematics from the birth of numbers. New York: W. W. Norton. pp. 963–965. ISBN 0-393-04002-X.
^ S.D. Poisson, Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés (Paris, France: Bachelier, 1837), page 206.
^ ^a ^b ^c Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p157
^ Ladislaus von Bortkiewicz, Das Gesetz der kleinen Zahlen [The law of small numbers] (Leipzig, Germany: B.G. Teubner, 1898). On page 1, Bortkiewicz presents the Poisson distribution. On pages 23–25, Bortkiewicz presents his famous analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preussischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.).
^ Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, Roy D. Yates, David Goodman, page 60.
^ For the proof, see : Proof wiki: expectation and Proof wiki: variance
^ Choi KP (1994) On the medians of Gamma distributions and an equation of Ramanujan. Proc Amer Math Soc 121 (1) 245–251
^ Riordan, John (1937). "Moment recurrence relations for binomial, Poisson and hypergeometric frequency distributions". Annals of Mathematical Statistics 8: 103–111. Also see Haight (1967), p. 6.
^ E. L. Lehmann (1986). Testing Statistical Hypotheses (second ed.). New York: Springer Verlag. ISBN 0-387-94919-4. page 65.
^ Raikov, D. (1937). On the decomposition of Poisson laws. Comptes Rendus (Doklady) de l' Academie des Sciences de l'URSS, 14, 9–11. (The proof is also given in von Mises, Richard (1964). Mathematical Theory of Probability and Statistics. New York: Academic Press. )
^ Laha, R. G. and Rohatgi, V. K. Probability Theory. New York: John Wiley & Sons. p. 233. ISBN 0-471-03262-X.
^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p159
^ Massimo Franceschetti and Olivier Dousse and David N. C. Tse and Patrick Thiran (2007). "Closing the Gap in the Capacity of Wireless Networks Via Percolation Theory". IEEE Transactions on Information Theory 53 (3): 1009–1018.
^ NIST/SEMATECH, '6.3.3.1. Counts Control Charts', e-Handbook of Statistical Methods, accessed 25 October 2006
^ Huiming, Zhang; Lili Chu, Yu Diao (2012). "Some Properties of the Generalized Stuttering Poisson Distribution and its Applications". Studies in Mathematical Sciences 5 (1): 11–26.
^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0-412-31760-5. page 196 gives the approximation and higher order terms.
^ ^a ^b Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p163
^ S. M. Ross (2007). Introduction to Probability Models (ninth ed.). Boston: Academic Press. ISBN 978-0-12-598062-3. pp. 307–308.
^ ^a ^b Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p171
^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p153
^ "The Poisson Process as a Model for a Diversity of Behavioural Phenomena"
^ Philip J. Boland. "A Biographical Glimpse of William Sealy Gosset". The American Statistician, Vol. 38, No. 3. (Aug., 1984), pp. 179-183. Retrieved 2011-06-22.
^ Aatish Bhatia. "What does randomness look like?". "Within a large area of London, the bombs weren’t being targeted. They rained down at random in a devastating, city-wide game of Russian roulette."
^ A. Colin Cameron, Pravin K. Trivedi (1998). Regression Analysis of Count Data. Retrieved 2013-01-30. "(p.5) The law of rare events states that the total number of events will follow, approximately, the Poisson distribution if an event may occur in any of a large number of trials but the probability of occurrence in any given trial is small."
^ Good, I. J. (1986). "Some statistical applications of Poisson's work". Statistical Science 1 (2): 157–180. doi:10.1214/ss/1177013690. JSTOR 2245435.
^ Paszek, Ewa. "Maximum Likelihood Estimation - Examples".
^ Garwood, F. (1936). "Fiducial Limits for the Poisson Distribution". Biometrika 28 (3/4): 437–442. doi:10.1093/biomet/28.3-4.437.
^ Breslow, NE; Day, NE (1987). Statistical Methods in Cancer Research: Volume 2—The Design and Analysis of Cohort Studies. Paris: International Agency for Research on Cancer. ISBN 978-92-832-0182-3.
^ Fink, Daniel (1997) A Compendium of Conjugate Priors
^ Loukas S, Kemp CD (1986) The index of dispersion test for the bivariate Poisson distribution. Biometrics 42(4) 941–948
^ P.X., Gallagher (1976). "On the distribution of primes in short intervals". Mathematika 23: 4–9.
^ "Some notes on the Poisson distribution"

References[edit]

Joachim H. Ahrens, Ulrich Dieter (1974). "Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions". Computing 12 (3): 223–246. doi:10.1007/BF02293108.
Joachim H. Ahrens, Ulrich Dieter (1982). "Computer Generation of Poisson Deviates". ACM Transactions on Mathematical Software 8 (2): 163–179. doi:10.1145/355993.355997.
Ronald J. Evans, J. Boersma, N. M. Blachman, A. A. Jagers (1988). "The Entropy of a Poisson Distribution: Problem 87-6". SIAM Review 30 (2): 314–317. doi:10.1137/1030059.
Donald E. Knuth (1969). Seminumerical Algorithms. The Art of Computer Programming, Volume 2. Addison Wesley.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

v t e Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

Contents