Suppose an urn contains N balls. Each ball has a unique number  \in \{1, 2, 3, ..., N \}. We randomly select n balls without replacement. Among the selected n balls, we are interested in the ball whose number is the kth smallest. Denote X = k^{th} smallest number among the n numbers of the randomly selected n balls.

(a) What is the probability mass function of X?

P(X = i) = \frac{\binom{i-1}{k-1}\binom{N-i}{n-k}}{\binom{N}{n}}

(b) Compute E[\frac{1}{X-1}] . (Assume k > 1)

\sum_{i=2}^{n} (\frac{1}{i-1})P(X = i) = \sum_{i=2}^{n} (\frac{1}{i-1}) \frac{\binom{i-1}{k-1}\binom{N-i}{n-k}}{\binom{N}{n}}...?

(c) Compute Var(X).