Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ and let $a>0$.

\textbf{Markov's inequality.} If $X\ge 0$ pointwise, then

with the \reftext{def:expectation-variance-2026a}{expectation} in $[0,\infty]$ (the inequality being trivial when the right side is infinite).

\textbf{Chebyshev's inequality.} If $X$ and $X^{2}$ have finite expectation, then

with the \reftext{def:expectation-variance-2026a}{variance} as defined there.