Poisson Sampling Distribution Generator

The defining characteristic of a Poisson distribution is that its mean and variance are identical. In a binomial sampling distribution, this condition is approximated as p becomes very small, providing that n is relatively large. In this event, the appropriate binomial probabilities can be approximated by way of the Poisson probability function

^{T}P_{(k out of n)} = | (e^{—m})(m^{k})k! |

e = | the base of the natural logarithms; and | |

m = | the mean of the Poisson distribution; |

The present page constructs the mean of the Poisson distribution as equal to np, the mean of the binomial distribution, and then calculates and graphically displays the Poisson probabilities for each of the "k out of n" integer outcomes at the lower end of the distribution, up to a maximum of k=2np. As the page opens, you will be prompted to enter the values of n and p. n must be an integer no smaller than 40 (preferably larger), and p must be a value no greater than .02 (preferably smaller). In general, the larger the value of n, the smaller should be the value of p. The value entered for p can be either a decimal fraction such as .001 or a common fraction such as 1/1000.

Note that these approximations are valid only in the degree that the binomial variance, npq, is a close approximation of the binomial mean. You will receive a red-letter message when the discrepancy between the two is greater than 2%.

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