ThmDex – An index of mathematical definitions, results, and conjectures.
F10945
Formulation 0
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}(X \geq \lambda) \leq \frac{1}{\varphi(\lambda)} \mathbb{E} \varphi(X) \end{equation}