Normal approximation of the binomial and of the Poisson distribution

Another important application of the theorem is that the binomial and the Poisson distribution can be approximated, for ``large numbers'', by a normal distribution. This is a general result, valid for all distributions which have the reproductive property under the sum. Distributions of this kind are the binomial, the Poisson and the $\chi^2$. Let us go into more detail:

The reproductive property of the binomial states that if $X_1$, $X_2$, $\ldots$, $X_m$ are $m$ independent variables, each following a binomial distribution of parameter $n_i$ and $p$, then their sum $Y=\sum_iX_i$ also follows a binomial distribution with parameters $n=\sum_i n_i$ and $p$. It is easy to be convinced of this property without any mathematics. Just think of what happens if one tosses bunches of three, of five and of ten coins, and then one considers the global result: a binomial with a large $n$ can then always be seen as a sum of many binomials with smaller $n_i$. The application of the central limit theorem is straightforward, apart from deciding when the convergence is acceptable. The parameters on which one has to base a judgment are in this case $\mu=n\,p$ and the complementary quantity $\mu^c=n\,(1-p)=n-\mu$. If they are both $\gtrsim 10$ then the approximation starts to be reasonable.

The same argument holds for the Poisson distribution. In this case the approximation starts to be reasonable when $\mu=\lambda \gtrsim 10$.