ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4741 on D1719: Expectation
Probabilistic Chebyshov's inequality for square function
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 $\lambda \in (0, \infty)$ be a D993: Real number.
Then \begin{equation} \mathbb{P}(X \geq \lambda) \leq \frac{1}{\lambda^2} \mathbb{E} X^2 \end{equation}
Proofs
Proof 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 $\lambda \in (0, \infty)$ be a D993: Real number.