This makes no sense to me. Why are we using the Poisson distribution to talk about uniform [0,1] random variables? Why do we do this complicated derivation just

This makes no sense to me. Why are we using the Poisson distribution to talk about uniform [0,1] random variables? Why do we do this complicated derivation just to show that Poisson(1;1) = 1/e, and why does this show the answer to our question? If someone could explain, that would be very helpful. There is a much clearer derivation that does not make use of the Poisson distribution here: http://mathworld.wolfram.com/UniformSumDistribution.html http://mathworld.wolfram.com/UniformSumDistribution.html (Starts at "Interestingly, ...".)