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# Zero Modified Poisson

### Input.

Input for the probability null occurences box Pr(r=0) has to be a proportion value between 0 and 1. Input for the mean or average Expected box can be any (decimal) value larger than zero. Input for the number Observed box must be an integer value, a whole positive number without decimals. Check Expected as Par. to set the expected as the poisson parameter instead of the mean. Check classical to get the well known poisson distribution.

### Explanation.

The zero modified Poisson distribution concerns a Poisson distribution where the probability of a zero outcome is arbitrarily set. For example, you want to know if the number of items people buy in a shop follows a Poisson distribution, then, when you check this at the cash register, you will see very few people with zero items. Enter this very low probability in the Pr(r=0) box, from the value one or more items in the shopping basket the expectations follow the usual poisson distribution. The zero truncated Poisson distribution is a special case and concerns a Poisson distribution without zeros. The zero truncated Poisson distribution can be used when you expect nobody at the cash register with zero items in their basket. Enter the value zero in the Pr(r=0) box for the zero truncated poisson distribution. The usual "classical" poisson distribution is also a special case of the zero modified distribution with the proportion expected in the zero box being e(-mean). Check the "classical" box for this distribution. If the expected probability of zero events is relatively high it is also known as the zero inflated poisson distribution.

Philosophically this distribution has exactly the same properties as the usual Poisson distribution and the user is therefore for further information referred to the Poisson discussion.

However, in an important way the zero modified poisson distribution is different from the usual Poisson. Whereas in the usual Poisson the distribution parameter, the distribution mean, and the distribution variance all have the same value, in the zero modified Poisson they are all different. If you input the mean expected count (the average number of items you expect in peoples shopping baskets) the program will echo the parameter, the variance and part of the distribution. The users who prefer to use the distribution parameter can check the "Expected as Par." box. The program will echo the zero modified mean and the rest. The relationship between the zero truncated mean E(x>=1) and the underlying full poisson mean (or zero truncated poisson parameter) E(x>=0) is given by: E(x>=0) = LambertW (- exp(-E(x>=1)) * E(x>=1) ) + E(x>=1) . The zero modified model is an extension of this zero truncated model. The full poisson mean determines the relation between probability pr(x) and the next probability pr(x+1). For large values the zero modified Poisson distribution parameter, mean and variance are similar. Also, in the case of "classical" poisson the distribution parameter, mean and variance of the mean are equal. An assumption of the poisson is that the probability of the next event is independent and constant with regard to the number of events already seen.

The Zero Truncated Poisson distribution is also known as the conditional poisson or the positive poisson distribution.

Please study the poisson distribution further by using the Poisson spreadsheet

### Further Reading.

Uitenbroek DG.The mathematical relationship between the number of events in which people are injured and the number of people injured. British Journal of Sports Medicine 1995: 126-128.->SISA

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