ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2016 on D1719: Expectation
Subresult of R1236: Markov's inequality
Probabilistic Markov's inequality

Let $X \in \text{Random}[0, \infty]$ be a D5452: Random unsigned real number.
Let $\lambda > 0$ be a D993: Real number.
Then $$\mathbb{P}(X \geq \lambda) \leq \frac{1}{\lambda} \mathbb{E}(X)$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X : \Omega \to [0, \infty]$ is a D4381: Random basic number on $P$
Let $\lambda > 0$ be a D993: Real number.
Then $$\mathbb{P}(X \geq \lambda) \leq \frac{1}{\lambda} \mathbb{E}_{\mathbb{P}} (X)$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X : \Omega \to [0, \infty]$ is a D4381: Random basic number on $P$
Let $\lambda > 0$ be a D993: Real number.
Then $$\mathbb{P}(X \geq \lambda) \leq \frac{1}{\lambda} \int_{\Omega} X(\omega) \, \mathbb{P}(d \omega)$$
Subresults
 ▶ R4132: Strict version of probabilistic Markov inequality ▶ R4131: Markov lower bound on unsigned basic expectation ▶ R4141: Probabilistic Chernoff inequality
Proofs
Proof 0
Let $X \in \text{Random}[0, \infty]$ be a D5452: Random unsigned real number.
Let $\lambda > 0$ be a D993: Real number.
This result is a particular case of R1236: Markov's inequality. $\square$