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

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

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

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

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

(c) Compute $\displaystyle Var(X)$.