ThmDex – An index of mathematical definitions, results, and conjectures.
Gaussian approximation to binomial distribution
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \text{Bernoulli}(\theta)$ each be a D207: Bernoulli random boolean number such that
(i) $X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
Then \begin{equation} \sum_{n = 1}^N \frac{X_n - \theta}{\sqrt{\theta (1 - \theta) N}} \overset{d}{\longrightarrow} \text{Gaussian}(0, 1) \quad \text{ as } \quad N \to \infty \end{equation}
Proofs
Proof 0
Let $X_1, X_2, X_3, \ldots \in \text{Bernoulli}(\theta)$ each be a D207: Bernoulli random boolean number such that
(i) $X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
Results
(i) R2484: Moments of a Bernoulli random boolean number
(ii) R2483: Variance of Bernoulli random boolean number

show that $\mathbb{E} X_1 = \theta$ and $\text{Var} X_1 = \theta (1 - \theta)$, whence this result is a consequence of R3843: I.I.D. real central limit theorem with the identity index sequence. $\square$