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Subject:

Sampling query

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Date:

Thu, 15 Apr 1999 01:10:47 +0100

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 I have the following query related to probability distributions which arise from sampling. I imagine that I have a population of N elements and I have r successes. Firstly I disribute the r successes randomly among the N elements in two fashion - with replacement and without replacement. If I distribute them without replacemnt then I randomly pick an element and attribute the proporty of success to it. I do not replace replace this element and I select another etc. When I allocate the successes without replacement then each element has either 1 success or 0 successes attributed to it. With replacement, an element can have more than 1 success attributed to it. Let me denote these two schemes as pop with rep and pop without rep. Now I imagine that I take a sample n from the population and count the number of successes. I can take the sample with replacement or without replacement. I am interested in the probability distribution of the number of successes. With pop without rep and a sample taken with replacement the probability distribution is just Binomial (n,p=1/r) With pop without rep and a sample taken without replacement the probability distribution is hypergeometric. With pop with rep (multiple successes possible per element) and sampling with replacement, I find that the distribution of successes for a particular element is binomial and the problem becomes a messy one of finding all the possible combinations. For instance a total number of successes of 3 could be made up of three seperate elements each with 1 success or 2 elements one with 2 successes the other with 1 or one element with 3 successes. With pop with rep and sampling without replacement I am stuck. Can anyone point me in the right direction. I feel that the last two examples I have described must be quite common in applications and I would welcome any advice on possible approximations or references. Thanks Glenn Treacy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%