Consider a loop of string with unit length. Take n cuts independently and randomly along the string, what is the expected length of the smallest and the largest piece?

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Vin Sen Lee 2016-2-8

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评论： Vin Sen Lee 2016-2-10

This is what I did.

The probability is (1+(1-n)x)^n

So, expected value of x is it integral for x varies from 0 to 1/n which evaluates to 1/n^2

If this is right how should I write the code?

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Guillaume 2016-2-9

编辑：Guillaume 2016-2-9

I think Jan's point was that a loop, having no start and end point, can't really have a length. Perimeter might be a more appropriate term.

Vin Sen Lee 2016-2-10

So, i just got clarification on this problem. My task is to write a function whose input is a number of cuts and output is the 1x2 array of the form [xmin xmax] where xmin is your experimental expected value of the smallest cut and xmax is your experimental expected value of the largest cut.

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Are Mjaavatten 2016-2-8

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Your question is not very clear. The code below is an answer to: How can I code a test of this result?

 N=100000;  % Number of samples
 n=8;       % Number of cuts
 d = zeros(N,n); % Allocate space for results
 for i = 1:N
    a = sort(rand(1,n));    % Draw random cut poins and distribute them along the string
    b = [a(end)-1,a];       % Join ends
    d(i,:) = sort(diff(b)); % Sort the pieces by length
 end
 mean_lengths = mean(d);   % mean_lengths(i) is the mean length of the i'th shortest piece
 disp(mean_lengths);