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Re: a combinatorial question
Posted:
Apr 20, 2012 2:16 PM


On 20120419, analyst41@hotmail.com <analyst41@hotmail.com> wrote: > Although it seems elementary, I am not aware that standard textbooks > treat this problem.
> There is a universal set U of N distinct objects. A fixed subset S of > n distinct objects is chosen from it (0 < n < N).
> Another subset T of m (0 < m < N) distinct objects is then chosen from > U. The question is what is the probability distribution of the > cardinality of S intersection T. N may be considered to be infinity, > although m/N and n/N are not vanishingly small.
This is exactly the hypergeometric distribution, for finite N. That is usually given as taking a sample of size n from a population of size N for which m are the "marked" elements.
By making it the intersection of two random sets, one can see that the distribution is symmetric in m and n, which one can see by expanding the usual formula. But this argument does not require calculation, and shows why this symmetry occurs.
 This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)4946054 FAX: (765)4940558



