Poisson distribution

Some events are rather rare , they don't happen that often. For instance, car accidents are the exception rather than the rule. Still, over a period of time, we can say something about the nature of rare events. An example is the improvement of traffic safety, where the government wants to know wether seat belts reduce the number of death in car accidents. Here, the poisson distribution can be a usefull tool to answer question about benefits of seat belt use. Other phenomena that often follow a poisson distribution are death of infants, the number of misprints in a book, the number of customers arriving, and the number of activations of a geiger counter. The poisson distribution was derived by the french mathematician Poisson in 1837, and the first application was the describtion of the number of death by horse kicking in the prussian army (Bortkiewicz, 1898).

The poisson distribution is a mathematical rule that assigns probabilities to the number occurences. The only thing we have to know to specify the poisson distribution is the mean number of occurences. In the movie we start with a mean of 1/2 (see the picture on this page, we are driving slowly, and the probability of no accident (x = 0) is large. In time the movie increases the mean, we are driving faster, and the probability of no accident decreases dramatically. That is, if you drive very fast you will probably end up in a car crash.

The poisson distribution resembles the binomial distribution if the probability of an accident is very small . However, if we want to use the binomial distribution we have to know both the number of people who make it safely from A to B, and the number of people who have an accident while driving from A to B, whereas the number of accidents is sufficient for applying the poisson distribution. Thus, the poisson distribution is cheaper to use because the number of accidents is usually recorded by the police department, whereas the total number of drivers is not.
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