> [!theorem] > > Let $X$ be a [[Random Variable|random variable]], then for all $t \in \real$, > $ > \bp\bracs{X \ge t} \le \inf_{a > 0} \frac{\ev(e^{aX})}{e^{at}} > $ > *Proof*. Let $a > 0$, then by [[Markov's Inequality]], > $ > \bp\bracs{X \ge t} = \bp\bracs{e^{aX} \ge e^{at}} \le \frac{\ev(e^{aX})}{e^{at}} > $