Question

Let X and Y be independent random variables each of which attains any value between 1 and n with probability 1/n. Compute E(|X − Y|) and simplify your answer.

Answer 1

X and Y are independent and both attains a value between 1 and n with probability 1/n
Since:


, since X and Y are independent.


Now, each X and Y have n possibilities. That makes n x n = n² possible pairs.
The minimum value of |x-y| would be 0, in cases where both X and Y attain same value.
|x-y| = 0 would occur 'n' times
|x-y| = 1 would occur '2(n-1)' times, X can take any value between 1 and n-1, Y gets restricted to the ONLY number just bigger than x. So total n-1 possibilities, but since (x,y) is an ordered pair, same is the case when Y takes the values between 1 and n-1. So total 2(n-1) possibilities.
Simillarly,
|x-y| = 2 would occur '2(n-2)' times
...
|x-y| = n-1 would occur 2 times.




In general,

so,

In general,

so,