Thread: Another IRV question... Inversions

Hey guys. I have another question regarding IRV. I did several of them between this question and the last I posted, but this one is confusing to me.

Here's the question:
Let A[1 ... n] be an array of n distinct numbers.
If i < j and A[i] > A[j], then the pair (i,j) is called an inversion of A.
Suppose that each element of A is chosen uniformly at random from the range 1 through n.
Use indicator random variables to compute the expected number of inversions.

Ok, so I did some trial and error and I think the maximum number of inversions (I understand the whole inversion explanation) is 2n. I didn't try it to very high numbers though... So I'm not entirely confident on that.

Now, do I want to set up my to be 0 ... 2n ?

Then how do I calculated the expected value of , would it be to be reducing to ? I'm trying to do this by... Possibility * Probability.

But now I think I may have mixed this whole thing up so any help is really appreciated. We only had the one lecture ("crash course") on this, so I'm a bit overwhelmed, honestly. Thank you!

It's late and I didn't read everything you wrote, but I do know that.....

equals reduces to

Originally Posted by mander

Hey guys. I have another question regarding IRV. I did several of them between this question and the last I posted, but this one is confusing to me.

Here's the question:
Let A[1 ... n] be an array of n distinct numbers.
If i < j and A[i] > A[j], then the pair (i,j) is called an inversion of A.
Suppose that each element of A is chosen uniformly at random from the range 1 through n.
Use indicator random variables to compute the expected number of inversions.

Ok, so I did some trial and error and I think the maximum number of inversions (I understand the whole inversion explanation) is 2n. I didn't try it to very high numbers though... So I'm not entirely confident on that.

Now, do I want to set up my to be 0 ... 2n ?

Then how do I calculated the expected value of , would it be to be reducing to ? I'm trying to do this by... Possibility * Probability.

But now I think I may have mixed this whole thing up so any help is really appreciated. We only had the one lecture ("crash course") on this, so I'm a bit overwhelmed, honestly. Thank you!

Mander,

What has the max number of inversions got to do with the problem?

Here is a hint.

Define if (i,j) is an inversion, 0 otherwise, for . See if you can find ; this is also .

The total number of inversions is . You want to find its expected value, . By the theorem on the expected value of a sum, this is equal to .

It may help to know the number of pairs (i, j) with i < j; this is just the number of combinations of n objects taken 2 at a time.

Go for it.