Two-stage allocations and the double \(Q\)-function (Q1408524)

Summary: Let \(m+n\) particles be thrown randomly, independently of each other into \(N\) cells, using the following two-stage procedure. 1. The first \(m\) particles are allocated equiprobably, that is, the probability of a particle falling into any particular cell is \(1/N\). Let the \(i\)th cell contain \(m_i\) particles on completion. Then associate with this cell the probability \(a_i=m_i/m\) and withdraw the particles. 2. The other \(n\) particles are then allocated polynomially, that is, the probability of a particle falling into the \(i\)th cell is \(a_i\). Let \(\nu=\nu(m,N)\) be the number of the first particle that falls into a non-empty cell during the second stage. We give exact and asymptotic expressions for the expectation E\(\nu\).