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Unformatted text preview: Poisson distributions and counting statistics Mukund Vengalattore Introduction This note contains a brief description of the properties of a Poisson distribution. This process describes the statistics of a wide range of random, independent events that occur in many physical measurements. Examples of random variables governed by Poisson statistics include the number of customers walking into a restaurant within any given time interval, the number of times this website is accessed per minute and the number of hits on a Geiger counter due to a nearby radioactive sample. Of particular interest to us is the arrival statistics of photons incident on the surface of a receptor. This is yet another process well described by Poisson statistics. In the following, we will derive the probability distribution function that describes this process and conclude with some mathematical properties of the Poisson distribution. The Poisson distribution Consider a physical measurement that involves counting the number of instances N ( t ) t of some event that occurs within a given time interval t . To consider a concrete example, this event might be registering a photon on a detector. Assume that these events are not distributed evenly but rather, are characterized by some degree of randomness. In other words, if I were to count the number of events within many intervals each of duration t , I would measure a distribution of this number N ( t ) about some mean value N . I can then define an average rate of events given by = N/ t (1) The Poisson distribution describes the statistics of a random variable that obeys the following assumptions: The number of events...
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- Spring '08