Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 2 / 4F / i / Solution

For $t>0$, monotonicity gives $\{X\ge k\}=\{e^{tX}\ge e^{tk}\}$. Apply Markov inequality to the nonnegative variable $e^{tX}$:If the expectation is infinite, the bound remains true but uninformative. At $t=0$ the right side is one, so that endpoint follows directly. Minimizing over $t\ge 0$ gives the Chernoff bound.