ThmDex – An index of mathematical definitions, results, and conjectures.

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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)	$\begin{equation} \text{Var} X = \infty \end{equation}$

Also known as

Variance need not be finite if expectation is finite for random real number, Counterexample: random real number with finite expectation but infinite variance

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)	$\begin{equation} \forall \, n \in \{ 1, 2, 3, \ldots \} : \mathbb{P}(X = n) = \frac{1}{n^3} \zeta(3) \end{equation}$

The first claim is a consequence of the results

(i)	R5274: Probability mass partition for expectation of a random natural number
(ii)	R268: Basel problem
(iii)	R3414: Necessary and sufficient condition for convergence of real harmonic series

while the latter claim is a consequence of the results

(i)	R5275: Probability mass partition for natural number moment of a random natural number
(ii)	R2954: Standard real harmonic series diverges

$\square$