# proof of Markov’s inequality

Define

 $Y=\begin{cases}d&X\geq d\\ 0&\text{otherwise}\\ \end{cases}.$

Then $0\leq Y\leq X$. Additionally, it follows immediately from the definition that $Y$ is a random variable (i.e., that it is measurable). Computing the expected value of $Y$, we have that

 $\mathbb{E}[X]\geq\mathbb{E}[Y]=d\cdot\mathbb{P}\left\{X\geq d\right\},$

and the inequality follows.

Title proof of Markov’s inequality ProofOfMarkovsInequality 2013-03-22 12:47:42 2013-03-22 12:47:42 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 7 Andrea Ambrosio (7332) Proof msc 60A99