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

Signed expectation finite iff absolutely integrable random basic number

Formulation 0

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

Then \begin{equation} \mathbb{E} |X| < \infty \quad \iff \quad - \infty < \mathbb{E} X < \infty \end{equation}

Formulation 1

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

Then \begin{equation} \mathbb{E} |X| < \infty \quad \iff \quad \mathbb{E} X \in (-\infty, \infty) \end{equation}

Formulation 2

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

Then \begin{equation} \mathbb{E} |X| \in [0, \infty) \quad \iff \quad \mathbb{E} X \in (-\infty, \infty) \end{equation}