ThmDex – An index of mathematical definitions, results, and conjectures.
Gaussian approximation to standard binomial distribution
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \text{Bernoulli}(1/2)$ each be a D3999: Standard 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 - \frac{1}{2}}{\sqrt{N / 4}} \overset{d}{\longrightarrow} \text{Gaussian}(0, 1) \quad \text{ as } \quad N \to \infty \end{equation}
Proofs
Proof 0
Let $X_1, X_2, X_3, \ldots \in \text{Bernoulli}(1/2)$ each be a D3999: Standard Bernoulli random boolean number such that
(i) $X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
This result is a particular case of R3602: Gaussian approximation to binomial distribution. $\square$