(i) Establish that E(X) for the geometric random variable is 1/p and that V ar(X) = q/p2, where q = 1 - p.
(ii) Given that for a certain geometric random variable, P (X = 2) = 0.0475 and P (X = 10) = 0.0315, determine P (2 ≤ X ≤ 10).
(iii) The average chain length of a polymer produced in a batch reactor is given as 200 units, where chain length itself is known to be a geometric random variable. What fraction of the polymer product is expected to have chains longer than 200 units?