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Probability question

  • Thread starter talolard
  • Start date
  • #1
125
0

Homework Statement


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.

The Attempt at a Solution


I've given this over an hour of thougt. I have no idea how to aproach this.
Thanks
Tal
 

Answers and Replies

  • #2
cronxeh
Gold Member
961
10
I take it you did not bother simulating this in MatLab..

Code:
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
 
  • #3
125
0
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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