I have $n$ Poisson events which occur with parameter $\lambda$. What can I expect the lowest of these to be?
I'd be happy with any reasonable interpretation of the question, including "what is the expectation of the lowest", "what is the median of the lowest", "what is the average of the lowest", etc. In my particular application (shot noise on low-light photos) $n$ is several million and $\lambda$ is at least in the hundreds, so a normal approximation would suit me just fine.