ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4777 on D5102: Basic expectation
Expectation of bounded random real number is within the bounding interval
Formulation 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $[a, b] \subset \mathbb{R}$ is a D544: Closed real interval
(ii) \begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation}
Then \begin{equation} \mathbb{E} X \in [a, b] \end{equation}
Formulation 1
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $[a, b] \subset \mathbb{R}$ is a D544: Closed real interval
(ii) \begin{equation} a \overset{a.s.}{\leq} X \overset{a.s.}{\leq} b \end{equation}
Then \begin{equation} \mathbb{E} X \in [a, b] \end{equation}
Proofs
Proof 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $[a, b] \subset \mathbb{R}$ is a D544: Closed real interval
(ii) \begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation}
Since $X \in [a, b]$ almost surely, result R4778: Stieltjes integral calculus expression for probability that a bounded random real number takes value on the bounding interval shows that $\int^b_a \, d F(x) = 1$. Thus, we have \begin{equation} \mathbb{E} X = \int^b_a x \, d F(x) \leq b \int^b_a \, d F(x) = b \end{equation} and \begin{equation} \mathbb{E} X = \int^b_a x \, d F(x) \geq a \int^b_a \, d F(x) = a \end{equation} This is what was required to be shown. $\square$