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

Result R4131 on D1719: Expectation

Markov lower bound on unsigned basic expectation

Formulation 0

Let $X \in \mathsf{Random}[0, \infty]$ be a D5101: Random unsigned basic number.
Let $\lambda > 0$ be a D993: Real number.

Then \begin{equation} \mathbb{E}(X) \geq \lambda \mathbb{P}(X \geq \lambda) \end{equation}

Proofs

Proof 0

Let $X \in \mathsf{Random}[0, \infty]$ be a D5101: Random unsigned basic number.
Let $\lambda > 0$ be a D993: Real number.

From R2016: Probabilistic Markov's inequality, we have the inequality \begin{equation} \frac{1}{\lambda} \mathbb{E}(X) \geq \mathbb{P}(X \geq \lambda) \end{equation} Multiplying both sides by $\lambda$, we then conclude \begin{equation} \mathbb{E}(X) \geq \lambda \mathbb{P}(X \geq \lambda) \end{equation} $\square$