Probability MT22, Markov's inequality
Flashcards
@State Markov’s inequality for a non-negative random variable $Y$ with finite expectation.
\[\forall t > 0, \quad \mathbb{P}(Y \ge t) \le \frac{\mathbb{E}[Y]}{t}\]
What random variable do you consider in the proof of Markov’s inequality?
\[\tilde{Y _ t} = \begin{cases} 0 &\text{if } Y < t \\ t &\text{if } Y \ge t \end{cases}\]
@Prove Markov’s inequality (∆markovs-inequality): let $Y$ be a non-negative random variable with finite expectation. Then for all $t > 0$,
\[\mathbb{P}(Y \ge t) \le \frac{\mathbb{E}[Y]}{t}\]
@todo (probability, page 64).