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

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

Result R4398 on D3161: Random real number

Formulation 0

Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \leq b$
(ii)	$[a, b] \subset \mathbb{R}$ is the D544: Closed real interval from $a$ to $b$
(iii)	$X \in \text{Random}[a, b]$ is a D3161: Random real number

Then

(1)	\begin{equation} \mathbb{E} X = a \quad \implies \quad X \overset{a.s.}{=} a \end{equation}
(2)	\begin{equation} \mathbb{E} X = b \quad \implies \quad X \overset{a.s.}{=} b \end{equation}

Formulation 1

Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \leq b$
(ii)	$[a, b] \subset \mathbb{R}$ is the D544: Closed real interval from $a$ to $b$
(iii)	$X \in \text{Random}[a, b]$ is a D3161: Random real number

Then

(1)	\begin{equation} \mathbb{E} X = a \quad \implies \quad \mathbb{P}(X = a) = 1 \end{equation}
(2)	\begin{equation} \mathbb{E} X = b \quad \implies \quad \mathbb{P}(X = b) = 1 \end{equation}