The expected number of inversions--From Introduction to Algorithms by Cormen

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. (See Problem 2-4 for more on inv

Answer 1

All the solutions seem to be correct, but the problem says that we should use indicator random variables. So here is my solution using the same:

    Let Eij be the event that i < j and A[i] > A[j].

    Let Xij = I{Eij} = {1 if (i, j) is an inversion of A

                        0 if (i, j) is not an inversion of A}

    Let X = Σ(i=1 to n)Σ(j=1 to n)(Xij) = No. of inversions of A.

    E[X] = E[Σ(i=1 to n)Σ(j=1 to n)(Xij)]

         = Σ(i=1 to n)Σ(j=1 to n)(E[Xij])

         = Σ(i=1 to n)Σ(j=1 to n)(P(Eij))

         = Σ(i=1 to n)Σ(j=i + 1 to n)(P(Eij)) (as we must have i < j)

         = Σ(i=1 to n)Σ(j=i + 1 to n)(1/2) (we can choose the two numbers in
                                            C(n, 2) ways and arrange them
                                            as required. So P(Eij) = C(n, 2) / n(n-1))

         = Σ(i=1 to n)((n - i)/2)

         = n(n - 1)/4