Thanks in advance for the help.

I'm working on a problem that would be greatly simplified if a solution exists for the following optimization problems for some $N \in \mathbb{N}^+$

$$min \underset{i \in N}{\prod} x_i \;\; s.t. \;\; y = \underset{i \in N}{\sum} x_i \; \wedge \; y,x_i \in [0,1]$$

$$max \underset{i \in N}{\prod} x_i \;\; s.t. \;\; y = \underset{i \in N}{\sum} x_i \; \wedge \; y,x_i \in [0,1]$$

At the moment the best I have been able to manage is randomly generating solutions (for my purposes $N$ tends to be small, less than 6 or 7). Does a solution for these problems exist? Alternatively, is there a trivial heuristic solution or do a set of tight bounds exist that I am just not seeing for some reason? I've noticed that for any given $y$ the interval of solutions tends to be fairly tight.

Note that I have included the statistics tag because of what $x_i$ represents in the problem I would like to use this for. Specifically, $x_i$ represents the probability of an event occurring and the product the probability of a specific outcome of random variable. The constraint I am using, a sum which can be larger than 1, is the result of heuristic I have been playing with that seems to work very well.


A bit more information for the eye brow raises I expect I'll get. I have a discrete random variable whose outcomes depend entirely on a set of events (ie $x_i$). Each $x_i$ is independent. What I'm interested in is the behavior of the probabilities of the set of events as they become more concentrated so to speak while the probability of a specific outcome remains constant. For example, with three events two possible solutions would be events with probabilities [.9,.9,.9] and [1,.729,1]. The second solution I would say is more concentrated than the second since the probability of the outcome I am interested in (ie all events occur) is based solely on whether the second event occurs. A natural statistic to use to measure this I believe would be the standard deviation, however, this does not appear to work very well. On a whim, I tried the summation of the events and found that for a specific summation the min and max values of $y$ tends to very tight which, while not perfect, gives me exactly what I want. However, while I know that the bound is tight via simulation I would prefer a closed form solution, if one exists. Also, I'm open to any other possible statistic that would capture ``concentration'' as abstract as this idea is (ideally for a concentration there would be exactly one $y$ value).


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