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Homework Help: Probability question

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Let [tex] x_1,x_2,...x_8 [/tex] be a random permutation of the set [/tex] {1,2,3,4,5,6,7,8} and [tex] S= (x_1-x_2)^2+(x_2-x_3)^2+(x_3-x_4)^2+...+(x_7-x_8)^2 [/tex]
    Calculate the expected value of S.

    3. The attempt at a solution
    I've given this over an hour of thougt. I have no idea how to aproach this.
    Thanks
    Tal
     
  2. jcsd
  3. Mar 8, 2010 #2

    cronxeh

    User Avatar
    Gold Member

    I take it you did not bother simulating this in MatLab..

    Code (Text):

    C = 0;
    for i=1:10000000
    X=randperm(8);
    S = ((X(1)-X(2))^2 + (X(2)-X(3))^2 + (X(3)-X(4))^2 + (X(4)-X(5))^2 + (X(5)-X(6))^2 + (X(6)-X(7))^2 + (X(7)-X(8))^2);
    C = C + S;
    end

    C/i
     
    ans =

    83.9955
     
  4. Mar 11, 2010 #3
    It's the evening before my exam and I finally figured it out. Heres an outline of a solution:

    first calculate [tex] E[(x-y)^2|x \neq y] [/tex]
    There are 8 ways for x=y out of 6 so all of the other events have a chance of 1/56.
    Drawing a joint distribution table it is easy to find the probobalities of the different values of x-y and then calculate the expected value of the above expression which is 12.

    Now notice that in a permutation [tex] x_i \neq y_j [/tex].
    We just calculated the expected value of [tex] (x_j-x_i)^2 [/tex] for a specific j and i.
    So lets make an indicator random variable L. the expected value of L is 12. But the sum we are looking for is the sum of 7 such indicators and since the expected value of a sum is the sum of the expected values the result is 7*12=84.
     
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