Solution
= Solution
The <Markov inequality> states that for a nonnegative <random variable> $Y$ and $a>0$,
$$
\Pr(Y\geq a)\leq\frac{\mathbb EY}{a}.
$$
For $t>0$, the event $X\geq x$ is the same as $e^{tX}\geq e^{tx}$. Applying Markov's inequality to the nonnegative variable $e^{tX}$ gives the <exponential Markov bound>
$$
\boxed{\Pr(X\geq x)\leq e^{-tx}M_X(t)}.
$$