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mistaken_id
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 Expected maximum value in [0,1]   « on: Mar 17th, 2009, 10:16pm » Quote Modify

Suppose N people choose some value (real number) from the interval (0,1). What is the expected value of the maximum?

Also what is the expected value of the second maximum?
 « Last Edit: Mar 17th, 2009, 10:17pm by mistaken_id » IP Logged
Ronno
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 Re: Expected maximum value in [0,1]   « Reply #1 on: Mar 17th, 2009, 11:29pm » Quote Modify

Assuming each person chooses their number independently with uniform probability, the expected maximum is N/(N+1) and the expected second maximum is (N-1)/(N+1). In fact, the expected ith minimum is  i/(N+1).
 « Last Edit: Mar 17th, 2009, 11:36pm by Ronno » IP Logged

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mistaken_id
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 Re: Expected maximum value in [0,1]   « Reply #2 on: Mar 18th, 2009, 8:16am » Quote Modify

Can you explain how??
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Ronno
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 Re: Expected maximum value in [0,1]   « Reply #3 on: Mar 18th, 2009, 8:37am » Quote Modify

The maximum having value x (0<x<1) means that one person chose a number in [x, x+dx] and the other N-1 chose numbers in (0, x). So, the probability density function of the maximum f(x)= x^(N-1)/(int 0...1 x^(N-1)dx)

So the expected value is int 0...1 xf(x)dx which comes out to be N/(N+1)

Intuitively, you would expect the numbers to be uniformly distributed in the interval and the N+1 gaps between them to be equal which also conforms to the result.
 « Last Edit: Mar 18th, 2009, 8:42am by Ronno » IP Logged

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mistaken_id
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 Re: Expected maximum value in [0,1]   « Reply #4 on: Mar 19th, 2009, 10:18am » Quote Modify

on Mar 18th, 2009, 8:37am, ronnodas wrote:
 So the expected value is int 0...1 xf(x)dx which comes out to be N/(N+1)

The integral of 0...1 xf(x)dx is 1/(N+1) rite?? How did you get N/N+1
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Obob
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 Re: Expected maximum value in [0,1]   « Reply #5 on: Mar 19th, 2009, 10:36am » Quote Modify

Check the definition of f(x) again.
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