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?

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?

[Walter Roberson]

The probability of what?

[Jan]

Does a loop have a length? Are the randomly distributed cuts set according to a equasl, normal Gauss , Lorenz, etc distribution?

[Walter Roberson]

"unit" length according to the title.

[Vin Sen Lee]

yes unit length and i think is randomly distributed

[Guillaume]

'randomly distributed' does not mean anything. You can have 'uniform' distribution, 'normal' distribution, 'log normal' distribution, and many more.

[Guillaume]

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]

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.

Answer 1

[Are Mjaavatten]

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);

[Vin Sen Lee]

how do i get the longest length then?

[Walter Roberson]

mean_lengths(end) is the mean of the longest.

The shortest out of all of the runs is min(d(:)) and the longest out of all of the runs is max(d(:)) (those might occur on different runs.)