ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4518 on D2140: Random real number variance
Equivalent characterisations of finiteness of variance for a random real number
Formulation 1
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number.
Then the following statements are equivalent
(1) \begin{equation} \text{Var} X < \infty \end{equation}
(2) \begin{equation} \exists \, \lambda \in \mathbb{R} : \mathbb{E}|X - \lambda|^2 < \infty \end{equation}
(3) \begin{equation} \mathbb{E} X^2 < \infty \end{equation}
(4) \begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}