Poisson distribution
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| Poisson distribution | |
|---|---|
| Name | Poisson distribution |
| Type | Discrete probability distribution |
| Support | {0,1,2,...} |
| Parameter | λ > 0 |
Poisson distribution The Poisson distribution models counts of rare events occurring independently in a fixed interval and arises in stochastic processes, queueing theory, and reliability studies. It connects to the work of historical figures in probability and statistics and appears in applied problems linked to institutions and projects across science and engineering. Researchers at universities and organizations use it alongside other distributions in analyses from particle physics to epidemiology.
Definition and parameters
The distribution is defined by a single rate parameter λ, named in contexts by practitioners at École Polytechnique and researchers influenced by Siméon Denis Poisson; λ represents the expected count per interval used by analysts at Harvard University, University of Cambridge, Massachusetts Institute of Technology, Stanford University and University of Oxford. Applied teams at CERN, NASA, Centers for Disease Control and Prevention, World Health Organization and National Institutes of Health often specify λ when modeling rare events, while economists at International Monetary Fund and demographers at United Nations may use related count models. In engineering contexts studied at Bell Labs, General Electric, Siemens, IBM and Boeing, λ is estimated from observed frequencies and links to design criteria at Federal Aviation Administration and standards bodies.
Properties and moments
Key properties include mean = λ and variance = λ, which statisticians at Royal Statistical Society, American Statistical Association, Institute of Mathematical Statistics, London School of Economics and Princeton University routinely exploit. Higher moments and cumulants are integer multiples of λ, a fact used in theoretical work at Institut Henri Poincaré, Mathematical Association of America, Courant Institute, Weizmann Institute of Science and KTH Royal Institute of Technology. The distribution is closed under addition for independent variables with parameters estimated by teams at National Institute of Standards and Technology and laboratories at Los Alamos National Laboratory and Oak Ridge National Laboratory. Moment-generating functions and characteristic functions are taught in courses at Yale University, Columbia University, University of Chicago, University of California, Berkeley and Caltech.
Probability mass function and derivations
The probability mass function p(k;λ) = e^{-λ} λ^k / k! is derived via limits of binomial models and Poisson process constructions developed in conjunction with work referenced by scholars at École Normale Supérieure, University of Göttingen, Princeton Plasma Physics Laboratory and historical academies like Académie des Sciences. Derivations via the Poisson limit theorem invoke sequences studied by mathematicians connected to Humboldt University of Berlin, Sorbonne University, University of Edinburgh, University of Vienna and Scuola Normale Superiore. Alternative derivations use stochastic processes from research at Bell Labs, AT&T Laboratories, MIT Lincoln Laboratory, Los Alamos National Laboratory and Argonne National Laboratory.
Applications and examples
Applications span diverse domains: counting decay events in experiments at CERN and Fermilab; modeling call arrivals in systems designed by Bell Labs and Nokia; estimating defect counts in production at Toyota and Ford Motor Company; analyzing rare disease incidence in studies by World Health Organization, Centers for Disease Control and Prevention and university medical centers like Johns Hopkins University and Mayo Clinic. It appears in insurance risk models at Lloyd's of London and AIG, in telecommunications planning at Ericsson and AT&T, and in ecology work associated with Smithsonian Institution and Royal Botanic Gardens, Kew. Case studies by researchers at Imperial College London, University of Toronto, McGill University, University of Melbourne and Australian National University illustrate practical estimation and hypothesis testing.
Relation to other distributions
The distribution is the limit of the binomial distribution studied historically at University of Copenhagen and connects to the exponential and gamma distributions through Poisson process interarrival times examined at Princeton University and University of Michigan. Compound Poisson models are used in actuarial science at Society of Actuaries, Casualty Actuarial Society, Munich Re and Swiss Re. Mixtures and approximations involving negative binomial, normal and Skellam distributions are topics in monographs from Oxford University Press, Cambridge University Press, Springer and course material at Columbia University and University of California, Los Angeles.
Estimation and inference
Parameter estimation by maximum likelihood (λ̂ = sample mean) and methods-of-moments are standard in curricula at Royal Statistical Society, American Statistical Association, Institute of Actuaries and taught in programs at University of Pennsylvania, Cornell University, Duke University and University of Texas at Austin. Inference procedures, confidence intervals and hypothesis tests are implemented in software packages from The R Project for Statistical Computing, Python Software Foundation ecosystems used at Facebook, Google, Microsoft Research and applied in regulatory reporting for Securities and Exchange Commission and Food and Drug Administration.
Generalizations and extensions
Generalizations include nonhomogeneous Poisson processes researched at National Institute for Mathematical Sciences, spatial Poisson processes used by teams at National Geographic Society and Esri, and marked Poisson processes applied in seismology at United States Geological Survey and Japan Meteorological Agency. Extensions to zero-inflated and hurdle models are used in studies at Harvard School of Public Health, London School of Hygiene and Tropical Medicine, Johns Hopkins Bloomberg School of Public Health and actuarial research at Willis Towers Watson and Mercer.