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Re: a combinatorial question
Posted:
Apr 28, 2012 9:30 AM


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 = Nn 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 onesided tests for particular cells and groups of cells to test for over/under performance?



