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Negative Binomial

Input.

Input for the expected proportion, 'blue' box, should be a positive decimal value between '0' (zero) and '1' (one). One can not give the number expected. Input for the number 'blue' box must be an integer value, a whole positive number without decimals. Same for sample size, must be an integer value, a whole positive number without decimals.

A windows version of the negative binomial procedure is available here.

Explanation.

The negative binomial distribution is also known as the Pascal Distribution or the Polya Distribution. The Geometric Distribution is a special case of the negative binomial distribution with the number positive, in the second blue box, being '1' (one).

The negative binomial distribution does basically the same thing as the Binomial, except that now we are asking about the probability of a particular sample size, given that we have found 'x' results to be positive (or 'white', or 'car crashes'), whereas we had expected to find 'u' results to be positive. Input is the same as for the binomial, but now the output expresses a change in the number of cases and not, as in the binomial, the number found positive. It is not possible to fill in a number in the first (expected value) box; it has to be a proportion. The reason is that the bottom box (the number of cases) is not fixed and there is therefore no fixed relationship between the expected proportion and the expected number (the expected number will increase with the sample size, if the expected proportion stays the same).

If you are used to other discrete distributions, the output may come as a surprise. Usually there is a probability for a zero value: a phenomenon not being observed. This time the counting does not start at the '0' (zero) value but at the first meaningful value. The first meaningful value is that in which the sample size and the observed number of positive responses are equal. This is because you must have a sample size of at least as large as the number of positive responses observed.

Please study the negative binomial distribution further by using the negative binomial spreadsheet

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