De Moivre-Laplace and Poisson Limit Theorems

Abstract

We again consider a binomial distribution with probabilities p and q, — i.e. ${{p}_{k}} = \left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array} } \right){{p}^{k}}{{q}^{{n - k}}}{{p}_{k}}$ and, to fix ideas, we assume that p > q. The number k takes values from 0 to n, so we have n + 1 probabilities p _k . We now study the question of the behavior of these probabilities as a function of k, for large n. We will see that there is a relatively small domain of values of k (of size $\sqrt n$) where the p _k are comparatively large and a remaining domain where the p _k are negligible. In order to define those k for which p _k is large we find k ₀ such that p _{k 0} = max_k p _k . We have the relation:

$$\frac{{{{p}_{{k - 1}}}}}{{{{p}_{k}}}} = \frac{{\left( {\begin{array}{*{20}{c}} n \\ {k + 1} \\ \end{array} } \right)}}{{\begin{array}{*{20}{c}} n \\ k \\ \end{array} }}\frac{{{{p}^{{k + 1}}}{{q}^{{n - k - 1}}}}}{{{{p}^{k}}{{q}^{{n - k}}}}} = \frac{{n!k!\left( {n - k} \right)!}}{{\left( {k + 1} \right)!\left( {n - k - 1} \right)!n!q}}\frac{p}{q} = \frac{{\left( {n - k} \right)}}{{\left( {k + 1} \right)}}\frac{p}{q}$$

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