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

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analyst41@hotmail.com

Posts: 117
Registered: 4/27/05
Re: a combinatorial question
Posted: Apr 28, 2012 9:30 AM
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On Apr 22, 2:42 am, Lynne Vickson <clvick...@gmail.com> wrote:
> On Apr 19, 4:36 pm, "analys...@hotmail.com" <analys...@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.

>
> If N is finite and the choice of the m objects comprising T is
> "random", the cardinality of the intersection
> has a hypergeometric distribution. (The hypergeometric distribution
> gives the probability of k type 1 objects
> when m objects are chosen without replacement from a population of N1
> type 1 and N2 type 2 objects; in
> your problem, N1 = n, N2 = N-n and you are asking how many objects in
> the random set T are type 1.)  If
> N is "infinite" but n/N is nonzero, and if you pick a FINITE number m
> of objects, you now have the binomial limit of the
> hypergeometric, so the cardinality of T intersect S has the binomial
> distribution with parameters m and p = n/N.
>
> RGV


Thanks. I am looking at a contingency tables problem. Let x(i,j) =
observed count in row i and column j. r(i) = row sum of row i and
c(j) = column sum of column j and G = grand total count. Typically
r(i).c(j)/G is comapred to x(i,j) to test for interaction between rows
and columns. There seems to be an implied "binomial approximation"
here and now its clear exactly whats going on.

I have another question: Any particluar cell may over- or under-
perform with respect to the expected value under the null hypothesis
of no interaction. Are there one-sided tests for particular cells and
groups of cells to test for over/under performance?



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