Markov inequality

An upper bound on the tail probability of a nonnegative random variable using its expectation.

Markov inequality

Markov inequality: If $X$ is a nonnegative random variable and $a>0$ , then

\mathbb{P}(X \ge a) \;\le\; \frac{\mathbb{E}[X]}{a}.

This is a basic tool for bounding event probabilities using expectation . It directly implies the Chebyshev inequality (by applying it to $(X-\mathbb{E}[X])^2$ ) and is also a starting point for exponential tail bounds such as the Chernoff bound .