Markov’s Inequality

Suppose \(X\) is a real-valued random variable with non-negative support. \(\forall \alpha > 0\):

\[\mathbb{P}[X \geq \alpha] \leq \mathbb{E}[X]/\alpha\]

Equivalently:

\[\mathbb{P}[X \geq c\mathbb{E}[X]] \leq \frac{1}{c}\]

Intuitively, Markov’s inequality says that the probability a non-negative random variable exceeds more than \(c\) times its mean is inversely proportional to \(c\).

Comparison with other inequalities

Markov’s vs Chebyshev’s

Let \(X\) be a random variable such that \(\mathbb{P}[X=k] = \frac{c}{k^3}\) for \(k = 1, 2, 3,...\). Markov tells us that \(\mathbb{P}[X \geq t] \leq \frac{c \pi^2}{6t}\). Chebyshev tells us that \(\mathbb{P}[X \geq t] \leq \frac{\mathbb{V}[X]}{t^2} = \infty\) because \(\mathbb{V}[X] = \infty\). Cehbyshev’s doesn’t work because variance is infinite, but Markov is relable!