Calculating Expected Value of S for Probability Question | Tal

[talolard]

Homework Statement

Let x_1,x_2,...x_8 be a random permutation of the set [/tex] {1,2,3,4,5,6,7,8} and S= (x_1-x_2)^2+(x_2-x_3)^2+(x_3-x_4)^2+...+(x_7-x_8)^2
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 approach this.
Thanks
Tal

 

[cronxeh]

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

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

 

[talolard]

It's the evening before my exam and I finally figured it out. Heres an outline of a solution:

first calculate E[(x-y)^2|x \neq y]
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 x_i \neq y_j.
We just calculated the expected value of (x_j-x_i)^2 for a specific j and i.
So let's 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.