The DeMoivre-Laplace limit theorem

Definition:

• Special case of Central limit theorem of de Moivre-Laplace
• If $S_n$ denotes the number of success that occur when $n$ independent trials, each resulting in a success with probability $p$ (Binomial distribution), are performed, then, for any $a<b$, $P(a\le \frac{S_p-np}{\sqrt{np(1-p)}}\le b)\to \Phi (b)-\Phi (a)$
  • as $n\to \infty$, where:$\Phi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{-y^2/2}dy$