Let $X_1, \, \ldots, \, X_N \in \text{Gaussian}(\mu, \sigma)$ each be a
D210: Gaussian random real number such that
| (i) |
\begin{equation}
N \in \{ 2, 3, 4, \, \ldots \}
\end{equation}
|
| (ii) |
$X_1, \, \ldots, \, X_N$ is an D2713: Independent random collection
|
| (iii) |
\begin{equation}
\overline{X}
: = \frac{1}{N} \sum_{n = 1}^N X_n
\end{equation}
|
| (iv) |
\begin{equation}
S
: = \left( \frac{1}{N - 1} \sum_{n = 1}^N (X_n - \overline{X}_N)^2 \right)^{1 / 2}
\end{equation}
|
A
D3161: Random real number $T \in \text{Random}(\mathbb{R})$ is a
student's random real number with parameter $(N - 1, \mu, \sigma)$ if and only if
\begin{equation}
T
\overset{d}{=} \frac{\overline{X} - \mu}{S / \sqrt{N}}
\end{equation}
