Probability mass function

The horizontal axis is the index k, the number ofoccurrences. The function is only defined at integervalues of k. The connecting lines are only guides for

the eye.Cumulative distribution function

The horizontal axis is the index k, the number ofoccurrences. The CDF is discontinuous at the

integers of k and flat everywhere else because avariable that is Poisson distributed only takes on

integer values.NotationParameters > 0 (real)Support k { 0, 1, 2, 3, ... }pmf

Poisson

Poisson distributionFrom Wikipedia, the free encyclopedia

In probability theory and statistics, the Poissondistribution (French pronunciation [pwas ]; in Englishusually /pwsn/), named after French mathematicianSimon Denis Poisson, is a discrete probabilitydistribution that expresses the probability of a givennumber of events occurring in a fixed interval of timeand/or space if these events occur with a knownaverage rate and independently of the time since thelast event.[1] The Poisson distribution can also be usedfor the number of events in other specified intervalssuch as distance, area or volume.

For instance, suppose someone typically gets 4 piecesof mail per day on average. There will be, however, acertain spread: sometimes a little more, sometimes alittle fewer, once in a while nothing at all.[2] Given onlythe average rate, for a certain period of observation(pieces of mail per day, phonecalls per hour, etc.), andassuming that the process, or mix of processes, thatproduces the event flow is essentially random, thePoisson distribution specifies how likely it is that thecount will be 3, or 5, or 10, or any other number, duringone period of observation. That is, it predicts the degreeof spread around a known average rate of occurrence.[2]

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

Contents1 History2 Definition3 Properties

3.1 Mean3.2 Median3.3 Higher moments3.4 Other properties

4 Related distributions5 Occurrence

5.1 Derivation of Poisson distribution The law of rare events5.2 Multi-dimensional Poisson process5.3 Other applications in science

6 Generating Poisson-distributed randomvariables7 Parameter estimation

7.1 Maximum likelihood

CDF, or

(for , where is theincomplete gamma function and isthe floor function)

MeanMedianModeVarianceSkewnessEx.kurtosisEntropy

(for large )

MGFCFPGF

7.2 Confidence interval7.3 Bayesian inference7.4 Simultaneous estimation of multiplePoisson means

8 Bivariate Poisson distribution9 See also10 Notes11 References

HistoryThe distribution was first introduced by Simon DenisPoisson (17811840) and published, together with hisprobability theory, in 1837 in his work Recherches sur laprobabilit des jugements en matire criminelle et enmatire civile (Research on the Probability ofJudgments in Criminal and Civil Matters).[3] The workfocused on certain random variables N that count,among other things, the number of discrete occurrences(sometimes called "events" or arrivals) that take placeduring a time-interval of given length. The result hadbeen given previously by Abraham de Moivre (1711) inDe Mensura Sortis seu; de Probabilitate Eventuum inLudis a Casu Fortuito Pendentibus in PhilosophicalTransactions of the Royal Society, p. 219.[4]

A practical application of this distribution was made byLadislaus Bortkiewicz in 1898 when he was given thetask of investigating the number of soldiers in thePrussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to thefield of reliability engineering.[5]

DefinitionA discrete random variable X is said to have a Poisson distribution with parameter > 0, if, for k =0,1,2,,the probability mass function of X is given by:[6]

where

e is Euler's number (e = 2.71828...)k! is the factorial of k.

The positive real number is equal to the expected value of X and also to its variance[7]

The Poisson distribution can be applied to systems with a large number of possible events, each of which is

rare. How many such events will occur during a fixed time interval? Under the right circumstances, this is arandom number with a Poisson distribution.

PropertiesMean

The expected value of a Poisson-distributed random variable is equal to and so is its variance.The coefficient of variation is , while the index of dispersion is 1.[4]

The mean deviation about the mean is[4]

The mode of a Poisson-distributed random variable with non-integer is equal to , which is thelargest integer less than or equal to . This is also written as floor(). When is a positive integer, themodes are and 1.All of the cumulants of the Poisson distribution are equal to the expected value . The nth factorialmoment of the Poisson distribution is n.

Median

Bounds for the median () of the distribution are known and are sharp:[8]

Higher moments

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

where the {braces} denote Stirling numbers of the second kind.[9] The coefficients of the polynomialshave a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, thenDobinski'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 are independent, and , then

.

[10] 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.[11]

Other properties

The Poisson distributions are infinitely divisible probability distributions.[12][13]

The directed KullbackLeibler divergence of Pois(0) from Pois() is given by

Bounds for the tail probabilities of a Poisson random variable can be derived using aChernoff bound argument.[14]

Related distributionsIf and are independent, then the difference follows aSkellam distribution.If and are independent, then the distribution of conditional on

is a binomial distribution.

Specifically, given , .

More generally, if X1, X2,..., Xn are independent Poisson random variables with parameters 1, 2,..., n then

given . In fact,

.

If and the distribution of , conditional on X = k, is a binomial distribution,, then the distribution of Y follows a Poisson distribution

. In fact, if , conditional on X = k, follows a multinomial distribution,, then each follows an independent Poisson distribution

.

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number oftrials goes to infinity and the expected number of successes remains fixed see law of rare eventsbelow. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently largeand p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a goodapproximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, andan excellent approximation if n 100 and np 10.[15]

The Poisson distribution is a special case of generalized stuttering Poisson distribution (or stutteringPoisson distribution) with only a parameter.[16] The stuttering Poisson distribution can be deduced fromthe limiting distribution of univariate multinomial distribution. It is also a special case of a compoundPoisson distribution.For sufficiently large values of , (say >1000), the normal distribution with mean and variance

(standard deviation ), is an excellent approximation to the Poisson distribution. If is greater thanabout 10, then the normal distribution is a good approximation if an appropriate continuity correction isperformed, i.e., P(X x), where (lower-case) x is a non-negative integer, is replaced by P(X x + 0.5).

Variance-stabilizing transformation: When a variable is Poisson distributed, its square root isapproximately normally distributed with expected value of about and variance of about 1/4.[17][18]Under this transformation, the convergence to normality (as increases) is far faster than theuntransformed variable.[citation needed] Other, slightly more complicated, variance stabilizingtransformations are available,[18] 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 withmean t, then the sequence of inter-arrival times are independent and identically distributedexponential random variables having mean 1 / .[19]The cumulative distribution functions of the Poisson and chi-squared distributions are related in thefollowing ways:[20]

and[21]

OccurrenceApplications of the Poisson distribution can be found in many fields related to counting:[22]

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 ofdiscrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in agiven area) whenever the probability of the phenomenon happening is constant in time or space. Examples ofevents 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. Thisexample was made famous by a book of Ladislaus Josephovich Bortkiewicz (18681931).The number of yeast cells used when brewing Guinness beer. This example was made famous byWilliam Sealy Gosset (18761937).[23]The number of phone calls arriving at a call centre within a minute. This example was made famous byA.K. Erlang (1878 1929).Internet traffic.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.

Comparison of the Poisson distribution (black lines)and the binomial distribution with n=10 (red circles),n=20 (blue circles), n=1000 (green circles). Alldistributions have a mean of 5. The horizontal axisshows the number of events k. Notice that as ngets larger, the Poisson distribution becomes anincreasingly better approximation for the binomialdistribution with the same mean.

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.[24]

Gallagher in 1976 showed that the counts of prime numbers in short intervals obey a Poisson distributionprovided a certain version of an unproved conjecture of Hardy and Littlewood is true.[25]

Derivation of Poisson distribution The law of rare events

See also: Poisson limit theorem

The Poisson distribution may be derived by considering aninterval, in time, space or otherwise, in which eventshappen at random with a known average number . Theinterval is divided in subintervals of equalsize, such that > . The probability that an event will fallin the subinterval is for each equal to , and theoccurrence of an event in may be approximatelyconsidered to be a Bernoulli trial. The total number ofevents then will be approximately binomial distributed withparameters and The approximation will be betterwith increasing ; the -distribution converges tothe Poisson distribution with parameter in the limit as napproaches infinity.

In several of the above examplessuch as, the number ofmutations in a given sequence of DNAthe events beingcounted are actually the outcomes of discrete trials, andwould more precisely be modelled using the binomialdistribution, that is

In such cases n is very large and p is very small (and sothe expectation np is of intermediate magnitude). Then thedistribution may be approximated by the less cumbersomePoisson distribution[citation needed]

This approximation is sometimes known as the law of rare events,[26] since each of the n individual Bernoullievents rarely occurs. The name may be misleading because the total count of success events in a Poissonprocess need not be rare if the parameter np is not small. For example, the number of telephone calls to abusy switchboard in one hour follows a Poisson distribution with the events appearing frequent to theoperator, but they are rare from the point of view of the average member of the population who is veryunlikely to make a call to that switchboard in that hour.[citation needed]

The word law is sometimes used as a synonym of probability distribution, and convergence in law meansconvergence in distribution. Accordingly, the Poisson distribution is sometimes called the law of smallnumbers 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 LadislausBortkiewicz (Bortkevitch)[27] about the Poisson distribution, published in 1898. Some have suggested that thePoisson distribution should have been called the Bortkiewicz distribution.[28]

Multi-dimensional Poisson process

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 unidimensionalprocesses. Thus, if D is any region the multidimensional space for which |D|, the area or volume of theregion, is finite, and if N(D) is count of the number of events in D, then

Other applications in science

In a Poisson process, the number of observed occurrences fluctuates about its mean with a standarddeviation . These fluctuations are denoted as Poisson noise or (particularly in electronics) as shotnoise.[citation needed]

The correlation of the mean and standard deviation in counting independent discrete occurrences is usefulscientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contributionof a single occurrence, even if that contribution is too small to be detected directly. For example, the charge eon an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If Nelectrons pass a point in a given time t on the average, the mean current is ; since the currentfluctuations should be of the order (i.e., the standard deviation of the Poisson process), thecharge can be estimated from the ratio .[citation needed]

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

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

Generating Poisson-distributed random variablesA simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) hasbeen given by Knuth (see References below):

algorithm poisson random number (Knuth): init: Let L e, 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 the returned value k, which is on average. There are many otheralgorithms to overcome this. Some are given in Ahrens & Dieter, see References below. Also, for large valuesof , there may be numerical stability issues because of the term e. One solution for large values of isrejection sampling, another is to use a Gaussian approximation to the Poisson.Inverse transform sampling is simple and efficient for small values of , and requires only one uniformrandom number u per sample. Cumulative probabilities are examined in turn until one exceeds u.

Parameter estimation

Maximum likelihood

Given a sample of n measured values ki=0,1,2,..., i=1,...,n, we wish to estimate the value of the parameter of the Poisson population from which the sample was drawn. The maximum likelihood estimate is [29]

Since each observation has expectation so does this sample mean. Therefore the maximum likelihoodestimate is an unbiased estimator of . It is also an efficient estimator, i.e. its estimation variance achieves theCramrRao lower bound (CRLB).[citation needed] Hence it is MVUE. Also it can be proved that the sum (andhence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for .To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass functionof the joint Poisson distribution for the sample into two parts: one that depends solely on the sample (called ) and one that depends on the parameter and the sample only through the function .Then is a sufficient statistic for .

Note that the first term, , depends only on . The second term, , depends on the sample

only through . Thus, is sufficient.

For completeness, a family of distributions is said to be complete if and only if implies that

for all . If the individual are iid , then .

Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.

For this equality to hold, it is obvious that must be 0. This follows from the fact that none of the other

terms will be 0 for all in the sum and for all possible values of . Hence, for all implies that, and the statistic has been shown to be complete.

Confidence interval

The confidence interval for the mean of a Poisson distribution can be expressed using the relationshipbetween the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squareddistribution 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 , a confidence interval for with confidencelevel 1 is

or equivalently,

where is the quantile function (corresponding to a lower tail area p) of the chi-squared distributionwith n degrees of freedom and is the quantile function of a Gamma distribution with shapeparameter n and scale parameter 1.[20][30] This interval is 'exact' in the sense that its coverage probability isnever less than the nominal 1 .

When quantiles of the Gamma distribution are not available, an accurate approximation to this exact intervalhas been proposed (based on the WilsonHilferty transformation):[31]

where denotes the standard normal deviate with upper tail area / 2.

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

calculate an interval for =n, and then derive the interval for .

Bayesian inference

In Bayesian inference, the conjugate prior for the rate parameter of the Poisson distribution is the gammadistribution.[32] Let

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

Then, given the same sample of n measured values ki as before, and a prior of Gamma(, ), the posteriordistribution is

The posterior mean E[] approaches the maximum likelihood estimate in the limit as .

[citation needed]

The posterior predictive distribution for a single additional observation is a negative binomial distribution,[33]sometimes called a GammaPoisson distribution.

Simultaneous estimation of multiple Poisson means

Suppose is a set of independent random variables from a set of Poisson distributions,each with a parameter , , and we would like to estimate these parameters. Then, Clevenson

and Zidek[34] show that under the normalized squared error loss , when

, then, similar as in Stein's famous example for the Normal means, the MLE estimator isinadmissible.

In this case, a family of minimax estimators is given for any and as[35]

Bivariate Poisson distribution

This distribution has been extended to the bivariate case.[36] The generating function for this distribution is

with

The marginal distributions are Poisson(1) and Poisson(2) and the correlation coefficient is limited to therange

A simple way to generate a bivariate Poisson distribution is to take three independent Poissondistributions with means and then set . The probabilityfunction of the Bivariate Poisson distribution is

See alsoCompound Poisson distributionConwayMaxwellPoisson distributionErlang distributionHermite distributionIndex of dispersionNegative binomial distributionPoisson clumpingPoisson process

Poisson regressionPoisson samplingQueueing theoryRenewal theoryRobbins lemmaTweedie distributionsZero-inflated modelZero-truncated Poisson distribution

Notes^ Frank A. Haight (1967). Handbook of the Poisson Distribution. New York: John Wiley & Sons.1.^

a

b "Statistics | The Poisson Distribution" (http://www.umass.edu/wsp/statistics/lessons/poisson/index.html).

Umass.edu. 2007-08-24. Retrieved 2012-04-05.2.

^ S.D. Poisson, Probabilit des jugements en matire criminelle et en matire civile, prcdes des rglesgnrales du calcul des probabilitis (Paris, France: Bachelier, 1837), page 206 (http://books.google.com/books?id=uovoFE3gt2EC&pg=PA206#v=onepage&q&f=false).

3.

^ a

b

c Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN

0-471-54897-9, p1574.

^ Ladislaus von Bortkiewicz, Das Gesetz der kleinen Zahlen [The law of small numbers] (Leipzig, Germany: B.G.Teubner, 1898). On page 1 (http://books.google.com/books?id=o_k3AAAAMAAJ&pg=PA1#v=onepage&q&f=false),Bortkiewicz presents the Poisson distribution. On pages 2325 (http://books.google.com/books?id=o_k3AAAAMAAJ&pg=PA23#v=onepage&q&f=false), Bortkiewicz presents his famous analysis of "4.Beispiel: Die durch Schlag eines Pferdes im preussischen Heere Getteten." (4. Example: Those killed in thePrussian army by a horse's kick.).

5.

^ Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, Roy D.Yates, David Goodman, page 60.

6.

^ For the proof, see : Proof wiki: expectation (http://www.proofwiki.org/wiki/Expectation_of_Poisson_Distribution)and Proof wiki: variance (http://www.proofwiki.org/wiki/Variance_of_Poisson_Distribution)

7.

^ Choi KP (1994) On the medians of Gamma distributions and an equation of Ramanujan. Proc Amer Math Soc 121(1) 245251

8.

^ Riordan, John (1937). "Moment recurrence relations for binomial, Poisson and hypergeometric frequencydistributions". Annals of Mathematical Statistics 8: 103111. Also see Haight (1967), p. 6.

9.

^ E. L. Lehmann (1986). Testing Statistical Hypotheses (second ed.). New York: Springer Verlag.ISBN 0-387-94919-4. page 65.

10.

^ Raikov, D. (1937). On the decomposition of Poisson laws. Comptes Rendus (Doklady) de l' Academie desSciences de l'URSS, 14, 911. (The proof is also given in von Mises, Richard (1964). Mathematical Theory ofProbability and Statistics. New York: Academic Press.)

11.

^ Laha, R. G. and Rohatgi, V. K. Probability Theory. New York: John Wiley & Sons. p. 233. ISBN 0-471-03262-X.12.^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN0-471-54897-9, p159

13.

^ Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and ProbabilisticAnalysis. Cambridge University Press. p. 97. ISBN 0521835402.

14.

^ NIST/SEMATECH, '6.3.3.1. Counts Control Charts (http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm)', e-Handbook of Statistical Methods, accessed 25 October 2006

15.

^ Huiming, Zhang; Lili Chu, Yu Diao (2012). "Some Properties of the Generalized Stuttering Poisson Distribution andits Applications" (http://cscanada.net/index.php/sms/article/view/j.sms.1923845220120501.Z0697). Studies inMathematical Sciences 5 (1): 1126. doi:10.3968/j.sms.1923845220120501.Z0697 (http://dx.doi.org/10.3968%2Fj.sms.1923845220120501.Z0697).

16.

^ 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.

17.

^ a

b Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN

0-471-54897-9, p16318.

^ S. M. Ross (2007). Introduction to Probability Models (ninth ed.). Boston: Academic Press.ISBN 978-0-12-598062-3. pp. 307308.

19.

^ a

b Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN

0-471-54897-9, p17120.

^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN0-471-54897-9, p153

21.

^ "The Poisson Process as a Model for a Diversity of Behavioural Phenomena" (http://www.rasch.org/memo1963.pdf)

22.

^ Philip J. Boland. "A Biographical Glimpse of William Sealy Gosset" (http://wfsc.tamu.edu/faculty/tdewitt/biometry/Boland%20PJ%20(1984)%20American%20Statistician%2038%20179-183%20-%20A%20biographical%20glimpse%20of%20William%20Sealy%20Gosset.pdf). The American Statistician, Vol. 38, No. 3. (Aug., 1984), pp. 179-183.Retrieved 2011-06-22.

23.

^ Aatish Bhatia. "What does randomness look like?" (http://www.empiricalzeal.com/2012/12/21/what-does-randomness-look-like/). "Within a large area of London, the bombs werent being targeted. They rained down atrandom in a devastating, city-wide game of Russian roulette."

24.

^ P.X., Gallagher (1976). "On the distribution of primes in short intervals" (http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7266644). Mathematika 23: 49.

25.

^ A. Colin Cameron, Pravin K. Trivedi (1998). Regression Analysis of Count Data (http://books.google.com/books?id=SKUXe_PjtRMC&pg=PA5&dq=%22law+of+rare+events%22+poisson&hl=en&sa=X&ei=xocJUbnpKuHL0AG4nYHADQ&ved=0CDMQ6AEwAQ#v=onepage&q=%22law%20of%20rare%20events%22%20poisson&f=false). Retrieved 2013-01-30. "(p.5) The law of rare events states that the total number ofevents will follow, approximately, the Poisson distribution if an event may occur in any of a large number of trials butthe probability of occurrence in any given trial is small."

26.

^ Edgeworth, F. Y. (1913). "On the use of the theory of probabilities in statistics relating to society"(http://www.jstor.org/stable/10.2307/2340091). Journal of the Royal Statistical Society 76: 165193.

27.

^ Good, I. J. (1986). "Some statistical applications of Poisson's work". Statistical Science 1 (2): 157180.doi:10.1214/ss/1177013690 (http://dx.doi.org/10.1214%2Fss%2F1177013690). JSTOR 2245435 (//www.jstor.org/stable/2245435).

28.

^ Paszek, Ewa. "Maximum Likelihood Estimation - Examples" (http://cnx.org/content/m13500/latest/?collection=col10343/latest).

29.

^ Garwood, F. (1936). "Fiducial Limits for the Poisson Distribution". Biometrika 28 (3/4): 437442.doi:10.1093/biomet/28.3-4.437 (http://dx.doi.org/10.1093%2Fbiomet%2F28.3-4.437).

30.

^ Breslow, NE; Day, NE (1987). Statistical Methods in Cancer Research: Volume 2The Design and Analysis ofCohort Studies (http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php). Paris: International Agency forResearch on Cancer. ISBN 978-92-832-0182-3.

31.

^ Fink, Daniel (1997) A Compendium of Conjugate Priors32.^ Gelman et al., Bayesian Data Analysis, 2nd ed. (2005) p. 60.33.^ Clevenson ML, Zidek JV (1975) Simultaneous Estimation of the Means of Independent Poisson Laws. Journal ofthe American Statistical Association 70(351a)

34.

^ Berger JO (1985) Statistical Decision Theory and Bayesian Analysis, 2nd Edition. Springer35.^ Loukas S, Kemp CD (1986) The index of dispersion test for the bivariate Poisson distribution. Biometrics 42(4)941948

36.

ReferencesJoachim H. Ahrens, Ulrich Dieter (1974). "Computer Methods for Sampling from Gamma, Beta, Poissonand Binomial Distributions". Computing 12 (3): 223246. doi:10.1007/BF02293108 (http://dx.doi.org

/10.1007%2FBF02293108).Joachim H. Ahrens, Ulrich Dieter (1982). "Computer Generation of Poisson Deviates". ACMTransactions on Mathematical Software 8 (2): 163179. doi:10.1145/355993.355997 (http://dx.doi.org/10.1145%2F355993.355997).Ronald J. Evans, J. Boersma, N. M. Blachman, A. A. Jagers (1988). "The Entropy of a PoissonDistribution: Problem 87-6". SIAM Review 30 (2): 314317. doi:10.1137/1030059 (http://dx.doi.org/10.1137%2F1030059).Donald E. Knuth (1969). Seminumerical Algorithms. The Art of Computer Programming, Volume 2.Addison Wesley.

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