A Poisson approximation in an urn model with indistinguishable balls
It is considered the distribution of n indistinguishable balls into m urns when the possible arrangements are equally probable. Let \(M_ k\) be the number of urns containing exactly k balls. The probability \(P(M_ k=r)\) is approximated by the term \(p(r;\mu)=e^{-\mu}\mu^ r/r!\), where \(\mu =m^ 2n^ k/(m+n)^{k+1}\) and \(\mu\), m and n are chosen such that \(\mu^ 2/m\) and \(\mu^ 2/n\) tend to 0 as m and n tend to infinity. The remainder term is developed so that the rate of convergence can be estimated. The approximation of \(P(M_ k\leq r)\) is given in the same way. The used method is elementary.
scientific article; zbMATH DE number 4178231