Let $X_1, X_2, X_3, \ldots \in \text{Random} \{ -1, 1 \}$ each be a
D5075: Random integer such that
| (i) |
\begin{equation}
\forall \, n \in \{ 1, 2, 3, \ldots \}
: \mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = 1/2
\end{equation}
|
| (ii) |
$X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection
|
A
D3161: Random real number $Z \in \text{Random}(\mathbb{R})$ is a
standard gaussian random real number if and only if
\begin{equation}
Z
\overset{d}{=}
\lim_{N \to \infty} \left( \frac{X_1}{\sqrt{N}} + \frac{X_2}{\sqrt{N}} + \cdots + \frac{X_N}{\sqrt{N}} \right)
\end{equation}
