I have a problem that I need help on.
I want to prove that the de Moivre-Laplace Theorem is a special case of the Central Limit Theorem (note that a binomial random variable is a sum of independent Bernoulli trials). This is what I have tried:
Let $ X_{1},X_{2},\ldots,X_{n} $ be Bernoulli trials. If $ S_{n} \stackrel{\text{df}}{=} X_{1} + X_{2} + \cdots + X_{n} $, then $$ \mathbf{E}[S_{n}] = n \mu \qquad \text{and} \qquad \mathbf{Var}[S_{n}] = n \sigma^{2}, $$ where $ \mu $ and $ \sigma $ are, respectively, the mean and standard deviation of the $ X_{i} $’s.
Standardizing $ S_{n} $, we get $ Z_{n} = \dfrac{S_{n} - n \mu}{\sqrt{n} \sigma} $. However, $$ \mathbf{E}[S_{n}] = n p \qquad \text{and} \qquad \mathbf{Var}[S_{n}] = n p q, $$ so $ Z_{n} = \dfrac{S_{n} - n p}{\sqrt{n p q}} $. Applying the Central Limit Theorem, I get the de Moivre-Laplace Theorem.
Thanks for your help, and have a nice day.