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

Result R4876 on D3161: Random real number
Subresult of R4875: Popoviciu's inequality

Interval length upper bound to variance of bounded random real number

Formulation 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, a, b \in \mathbb{R} : \mathbb{P}(X \in [a, b]) = 1 \end{equation}

Then \begin{equation} \text{Var} X \leq (b - a)^2 \end{equation}

Proofs

Proof 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \exists \, a, b \in \mathbb{R} : \mathbb{P}(X \in [a, b]) = 1 \end{equation}

This result is an immediate consequence of R4875: Popoviciu's inequality. $\square$