 Solution

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.