I'm too lazy to do all the computations... just consider the double sum as fixing an index and separating the other sum according to the first, so that you can remove the absolute value.
It's the absolute value that throws me off. Any hints on what to do? I'd appreciate it!
Let X,Y be uniform random variables with state space {1,2,...,n} i.e for each k in {1,...,n}, P(X=k)=P(Y=k)= (1/n). Suppose that X and Y are independent. Show that
E|X-Y|= (n-1)(n+1)/3n
This is the hint we were given:
sum from k=1 to n-1 of k =n(n-1)/2
sum from k=1 to n-1 of k^2= n(n-1)(2n-1)/6