# Tagged: Chebyshev’s inequality

## Problem 759

(a) Let $X$ be a random variable that takes only non-negative values. Prove that for any $a > 0$,
$P(X \geq a) \leq \frac{E[X]}{a}.$ This inequality is called Markov’s inequality.

(b) Let $X$ be a random variable with finite mean $\mu$ and variance $\sigma^2$. Prove that for any $a >0$,
$P\left(|X – \mu| \geq a \right) \leq \frac{\sigma^2}{a^2}.$ This inequality is called Chebyshev’s inequality.