## How to Prove Markov’s Inequality and 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**.