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Tricky Uniform Distribution Problem
I'm trying to use game theory to evaluate the optimal number of shills for an auction house to employ.
I'm having much more trouble with a seemingly simply lemma than I thought I would: Given n points randomly chosen along the interval [0,1] (openness or closedness irrelevant), what is the expected value of the largest point? I can't come up with anything prettier than saying it is a sequence described by: c(n) = a(n)/(2^((2^n)-1)) where a(n) is defined recursively by a(1) = 1 and a(n) = (a(n-1))^2+(2^((2^n)-2)) c(1)=1/2 c(2)=5/8 c(3)=89/128 c(4)=24305/32468 ... Beat, brag, or variance? Or am I missing something simple? Note that I'd prefer to keep this as a run-of-the-mill equation I can do calculations with, not a recursively defined sequence. |
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