Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 3F (Probability)

The Markov inequality states that for a nonnegative random variable $Y$ and $a>0$,For $t>0$, the event $X\ge x$ is the same as $e^{tX}\ge e^{tx}$. Applying Markov's inequality to the nonnegative variable $e^{tX}$ gives the exponential Markov bound

Independence givesThis is the moment generating function of a Poisson distribution with mean $n$. Hence

The exponential Markov bound gives, for every $t>0$,The exponent is minimized when $e^t=2$, so take $t=\log 2$. Then