62. LetI mage denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion ofI mage and Image in the permutation ifI mage andI mage precedesI mage. For instance the permutation 4, 2, 1, 5, 3 has 5 inversions—(4, 2), (4, 1), (4, 3), (2, 1), (5, 3). Consider now a random permutation of

—in the sense that each of the n! permutations is equally likely to be chosen—and let N denote the number of inversions in this permutation. Also, let and note thatI mage.

(a) Show that Image are independent random variables.

(b) What is the distribution of Image

(c) Compute and Var(N).