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

Result R4741 on D1719: Expectation
Subresult of R4686: Probabilistic Chebyshov's inequality for an exponentiation function

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.