From the above argument, we can deduce that for a large sample size \(n\), \(Q_i\) can approximated by \(Poisson(np_i)\).
2. Poissonization. Assume that the sample size \(N\) is random and Poissoned (i.e \(N\sim Poisson\left(n\right)\) and \(N\) is independent of \(X_i\)). Then, the marginal distributions of the \(Q_i\)'s end up being Poisson by the following theorem.