Poisson Distribution
The Poisson distribution is a mathematical rule that assigns probabilities to the number of occurrences of a certain event. It is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time. The Poisson distribution is one of the most important in probability. It is usually written as: P(x) = (xe-)/x!, where P(x) is the probability that the outcome of the function will be x, and is the average number of occurrences in a specified interval, either of space or time. In a Poisson distribution the mean and variance are equal and can be described by: E(x) = Var(x) = .
There are four assumptions made in order to apply the Poisson distribution to a problem. First, it is assumed that the probability of observing a single event over a small interval, either of space or time, is approximately proportional to the size of that interval. Also it is assumed that the probability of two events occurring in the same narrow interval is negligible. Third, the probability of an event occurring within a certain interval does not change over different intervals but remains the same. Lastly, the probability of an event in one interval is independent of the probability of an event in any other interval that is not overlapping with the first. So probabilities of intervals are not linked to each other. Violation of either of the last two assumptions can lead to overdispersion or extra variation. Aside from these assumptions that are made in order to effectively apply the Poisson distribution, there are some empirical tests that can be preformed to determine if a distribution is a Poisson distribution. To apply these tests the time/area intervals for all of the data should be the same. The simplest test is to determine if the variance is roughly equal to the mean for the data. If this is the case then the data may fit a Poisson distribution. Also a histogram graphical representation of the Poisson data should be skewed to the right, though this skewness may become less pronounced if the mean is large.
The Poisson distribution is very similar to the binomial distribution if the probability of an event occurring is very small. In some ways it is superior to the binomial distribution. For a binomial distribution one must know both the number of successful events as well as the number of unsuccessful events, whereas in the Poisson distribution one needs to know only the mean number of successful occurrences of an event. But as mentioned before, the binomial distribution can be applied in situations where the probability of an event occurring can be within a wide range but the Poisson distribution can be applied only when the probability of an event occurring is relatively large. Both methods are important in the theory of sampling.
The Poisson distribution was formulated by French mathematician Simeon Denis Poisson in 1837. He published the distribution in a work entitled Research on the Probability of Opinions. This method was developed to calculate the probability of the success of trials in situations where the probability of success on any one trial is extremely low but where the number of trials is very large. The first application was by Ladislaus von Bortkiewicz, a German professor born in Russia, in 1898 to the description of the number of deaths by horse kicking in the Prussian army.
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