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I've just come across something rather strange, I believe, about the micro-canonical derivation of the BE-distribution (as well as the Boltzmann and FD-distributions).

See for example https://en.wikipedia.org/wiki/Bose–Einstein_statistics#Derivation_from_the_microcanonical_ensemble

The starting assumption is that the system is isolated from the surroundings and has a fixed energy, which I'll call E.

On the other hand an integral over the resulting distribution function taken from energy E as lower integration boundary to infinity as upper boundary is not zero. This seems to indicate that there is a certain non-zero probability that an individual particle in the system can have an energy higher than the total energy of the system.

Am I misunderstanding something, or is there a way to explain this?