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The Poisson distribution 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. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The Poisson distribution may is derived by considering an interval, in time, space or otherwise, in which events happen at random with a known average number. The interval is divided in n subintervals of equal size. The probability that an event will fall in the subinterval is for each k equal to λ/n and the occurrence of an event Ik may be approximately considered to be a Bernoulli trial.

In cases where n is very large and p is very small, the distribution may be approximated by the less cumbersome Poisson distribution. This approximation is sometimes known as the law of rare events since each of the n individual Bernoulli events rarely occurs. The word law in statistics is sometimes used as a synonym of probability distribution and convergence in law means convergence in distribution. 

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Brakes for a Model of Automobile Fail with a Poisson

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Statistics: Characteristic function, pmf, and variance

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Working with the Poisson Random Variable

A manuscript is sent to a typing firm consisting of typists A, B, and C. If it is typed by A, then the number of errors made is a Poisson random variable with mean 2.6; if typed by B, then the number of errors is a Poisson random variable with mean 3; and if typed by C, then it is a Poisson random variable with mean 3.4. Let X d