Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 7 / c / ii

Insert the Poisson distribution probability generating function into the preceding bound:For fixed $x>\lambda$, differentiate the exponent with respect to $s$. Its derivative is $\lambda -x/s$, and its second derivative is $x/s^2>0$, so its unique minimum occurs at $s=x/\lambda >1$. At that value,ConsequentlyThis Chernoff bound remains valid for noninteger positive thresholds $x>\lambda$; the event is then the same as exceeding or equalling its integer ceiling.