Markov's Inequality

Markov's inequality is a distribution-free tail bound. It says that if a random variable is non-negative, then its mean controls how large it can be.

Statement

If $X \ge 0$ and $a>0$:

Intuition

If the average of $X$ is small, then $X$ cannot be “large” too often — otherwise the mean would be forced upward.

Markov is often the first stepping stone to stronger bounds:

• Apply Markov to $(X-\mu)^2$ to get Chebyshev’s inequality.

When it’s useful

• When you only know $E[X]$ (mean) and nothing else about the distribution
• For quick, conservative guarantees
• In proofs (building block for Chebyshev and concentration results)

Tightness (why it can be loose)

Markov can be conservative because it uses only the mean. If the distribution has heavy tails or lots of mass near 0, the bound may be far from the true probability.

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