Let $\mu>0$, $k\in\mathbb N^+$, and consider a random variable $Y\sim\mbox{Poisson}(\mu)$.
I am interested in (upper) bounding $\mathbb E[Y \mid Y \ge k]$, ideally with a closed-form expression.
Notice that we have that $\mathbb E[Y \mid Y \ge k]=\frac{\sum_{i=k+1}^{\infty} i\cdot \Pr[Y=i]}{\Pr[Y >k]}$. I have seen many different methods for upper bounding $\Pr[Y >k]$, however, this does not help in getting an upper bound for $\mathbb E[Y \mid Y \ge k]$.
Any ideas?