ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3681 on D3161: Random real number
Mean-deviance standardisation of a random real number
Formulation 1
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) \begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}
(ii) \begin{equation} \mu : = \mathbb{E} X \end{equation}
(iii) \begin{equation} \sigma : = \text{Std} X \end{equation}
Then
(1) \begin{equation} \mathbb{E} \left( \frac{X - \mu}{\sigma} \right) = 0 \end{equation}
(2) \begin{equation} \text{Var} \left( \frac{X - \mu}{\sigma} \right) = 1 \end{equation}