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?