Suppose you were looking at the Poisson distribution P(L, I) for a given Lambda and Index. How would you go about minimizing P to some arbitrary confidence level? My initial thought is to simply iterate away from Lambda (using a binary partition for example) until P is sufficiently minimal. Maybe not too inefficient, but surely there is a more direct approach for this?

In case it's not too clear, let me give an example. Let's say you own an online store and sell an average of 12 units per day. What is the most number of units you can expect to sell on any given day, within a certain measure of confidence? Well the chances of selling exactly 12 of them is roughly 11.5%. At 16 the probability drops down to ~7%, and at 18 units per day it falls to less than 3%. So all things being equal, ~97% of the time we should expect to sell no more than 18 units on any given day. So as long as we keep at least that many units in inventory, the chances of running out of stock should be pretty slim.

Can this "maximal minima" be calculated directly, and if so, how?