Let $X_1, X_2, X_3, \ldots \in \text{Rademacher} (1 / 2)$ each be a D5286: Rademacher random integer such that

(i)	$X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection

Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(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}

Let $X_1, X_2, X_3, \ldots \in \text{Random} \{ -1, 1 \}$ each be a D5075: Random integer such that

(i)	$X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
(ii)	\begin{equation} \forall \, n \in 1, 2, 3, \ldots : \mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2} \end{equation}

Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)

$a < b$

Then \begin{equation} \lim_{N \to \infty} \mathbb{P} \left( a \leq \frac{1}{\sqrt{N}} \sum_{n = 1}^N X_n \leq b \right) = \frac{1}{\sqrt{2 \pi}} \int^b_a e^{-x^2/2} \, d x \end{equation}