Let $X_1, X_2, X_3, \ldots \in \text{Rademacher} (1 / 2)$ each be a D5286: Rademacher random integer such that
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that
(i) | $X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection |
(i) | $a < b$ |
Then
\begin{equation}
\mathbb{P} \left( a \leq \frac{1}{\sqrt{N}} \sum_{n = 1}^N X_n \leq b \right) \longrightarrow \frac{1}{\sqrt{2 \pi}} \int^b_a e^{-x^2/2} \, d x
\quad \text{ as } \quad
N \to \infty
\end{equation}