Chernoff bound - Knowlpedia

Chernoff bound: Let $X$ be a random variable. Suppose $\mathbb{E}[e^{tX}]<\infty$ for some $t>0$ . Then for every real $a$ and every $t>0$ for which the expectation is finite,

\mathbb{P}(X\ge a)\le e^{-ta}\,\mathbb{E}\!\left[e^{tX}\right].

Similarly, if $\mathbb{E}[e^{-tX}]<\infty$ for some $t>0$ , then for every real $a$ and every such $t$ ,

\mathbb{P}(X\le a)\le e^{ta}\,\mathbb{E}\!\left[e^{-tX}\right].

Equivalent characterizations

Equivalently,

\mathbb{P}(X\ge a)\le \inf_{t>0} \; e^{-ta}\,\mathbb{E}\!\left[e^{tX}\right].

Remarks

This bound is obtained by applying Markov's inequality to the nonnegative random variable $e^{tX}$ , and it is naturally expressed in terms of the moment generating function of $X$ .