ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5273 on D6073: Basel random natural number
Finiteness of expectation in general does not imply finiteness of variance for random real number
Formulation 0
Let $X \in \text{Random}(\mathbb{N})$ be a D5216: Random natural number such that
(i) \begin{equation} \zeta(3) : = \sum_{n = 1}^{\infty} \frac{1}{n^3} \end{equation}
(ii) \begin{equation} \forall \, n \in \{ 1, 2, 3, \ldots \} : \mathbb{P}(X = n) = \frac{1}{n^3} \zeta(3) \end{equation}
Then
(1) \begin{equation} \mathbb{E} X = \frac{\pi^2}{6} \zeta(3) < \infty \end{equation}
(2) \begin{equation} \text{Var} X = \infty \end{equation}
Proofs
Proof 0
Let $X \in \text{Random}(\mathbb{N})$ be a D5216: Random natural number such that
(i) \begin{equation} \zeta(3) : = \sum_{n = 1}^{\infty} \frac{1}{n^3} \end{equation}
(ii) \begin{equation} \forall \, n \in \{ 1, 2, 3, \ldots \} : \mathbb{P}(X = n) = \frac{1}{n^3} \zeta(3) \end{equation}