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Topic: a combinatorial question
Replies: 4   Last Post: May 3, 2012 3:00 PM

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: a combinatorial question
Posted: Apr 20, 2012 2:16 PM
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On 2012-04-19, <> 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 Phone: (765)494-6054 FAX: (765)494-0558

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