ThmDex – An index of mathematical definitions, results, and conjectures.
F10946
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to [0, \infty]$ is a D5101: Random unsigned basic number on $P$
Let $\varphi : [0, \infty] \to \mathbb{R}$ be a D5321: Standard-isotone basic real function such that
(i) $\lambda > 0$ is a D993: Real number
(ii) \begin{equation} \varphi(\lambda) \neq 0 \end{equation}
Then \begin{equation} \mathbb{P} (\{ \omega \in \Omega : X(\omega) \geq \lambda \}) \leq \frac{1}{\varphi(\lambda)} \int_{\Omega} \varphi(X) \, d \mathbb{P} \end{equation}