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# Poisson

### Input.

Input for the mean or average Expected box can be any positive (decimal) value. Input for the number Observed box must be an integer value, a whole positive number without decimals.

Invert. To find the expectation which produces a certain cumulative probability value given the observed number. Give the probability value, a value between 0 and 1 in the Expected box. Put the observed number of cases in the Observed box, is an integer.

### Explanation.

The Poisson distribution has two applications:

1) The poisson distribution can be used as an alternative to the Binomial distribution in the case of very large samples. Your hypothesis was that you would find 'x' (the top cell) occurrences of a phenomenon, whereas in fact you found 'n' (the second input cell). The phenomenon might be the number of cases of a rare disease, the number of accidents on a busy junction, the number of stoppages on a production line, or the number of ethnic children at a football match. You want to test the assumption that environmental and other factors influencing the phenomenon were constant between observation periods. The program echoes the point probability and the probability of there being 'n' or more occurrences of a phenomenon given your expectation of 'x' occurrences. For the poisson distribution you do not need to give a sample size. If the sample size is known, it is generally preferable to use the Binomial.

The main differences between the poisson distribution and the binomial distribution is that in the binomial all eligible phenomena are studied, whereas in the poisson distribution only the cases with a particular outcome are studied. For example: in the binomial all cars are studied to see whether they have had an accident or not, whereas using the poisson distribution only the cars which have had an accidents are studied.

2) The poisson distribution can also be used to study how 'accidents' or 'malfunctions' or the chance of winning the lottery never, once or more than once, are distributed on the level of a population. If having one 'accident' has no influence on the chance of having another accident, the victim is 'put back into the population' immediately after an 'event', people may have one, two, three, or more accidents during a certain period of time. The Poisson distribution tells you how these chances are distributed. Mean or incidence is the number of accidents divided by the size of the population and is given to the program in the top expectation box. Note that although your calculation may result in a value between zero and one, this value is not a proportion but a true mean. You would get a true proportion if you divide the number of people who had an accident by the number of people (For a discussion of the relationships between these numbers see Uitenbroek 1995). In the second 'observed' box is given the number of accidents you want to study. If you give 10 in the observed box the output gives you the proportion of the population who had '0' (zero) accidents, the proportion who had '1' (one) accident, the proportion who had '2' (two) accidents etc. The cumulative distribution tells you the proportion that had '1' or more accidents, '2' or more etc.

The proportion of failures, thus the proportion in a population with zero events or accidents, is mathematically related to the Poisson mean as follows: Proportion(Failures) = exp (-Mean(poisson)); and then: Mean(poisson) = ln (1/Proportion(Failures)).

One assumption in this application of the poisson distribution is that the chance of having an accident is randomly distributed: every individual has an equal chance. Mathematically this is expressed in the fact that the variance and the mean for the poisson distribution are equal. A good way to check if this assumption that individuals have an equal chance of having the trait is correct, is to compare the variance of an (accident) distribution with its mean. If the variance is larger, then the assumption was not correct. The Negative Binomial Version 1 has been implemented to provide an alternative for the poisson distribution in the case of a non-random distribution.

Confidence intervals for a poisson estimate can be obtained by use of the SMR-Exact procedure. 