Let us model the chess tournament between Fisher and Spassky as a stochastic process. Let Xi, for i ≥ 1, be the duration of the ith game and assume that {Xi; i≥1} is a set of IID exponentially distributed rv s each with density fX (x) = λe-λx. Suppose that each game (independently of all other games, and independently of the length of the games) is won by Fisher with probability p, by Spassky with probability q, and is a draw with probability 1 - p - q. The first player to win n games is defined to be the winner, but we consider the match up to the point of winning as being embedded in an unending sequence of games.
(a) Find the distribution of time, from the beginning of the match, until the completion of the first game that is won (i.e., that is not a draw). Characterize the process of the number {N(t); t > 0} of games won up to and including time t. Characterize the process of the number {NF (t); t ≥ 0} of games won by Fisher and the number {NS(t); t ≥ 0} won by Spassky.
(b) For the remainder of the problem, assume that the probability of a draw is zero, i.e., that p + q = 1. How many of the first 2n - 1 games must be won by Fisher in order to win the match?
(c) What is the probability that Fisher wins the match? Your answer should not involve any integrals. Hint: Consider the unending sequence of games and use (b).
(d) Let T be the epoch at which the match is completed (i.e., either Fisher or Spassky wins). Find the CDF of T.
(e) Find the probability that Fisher wins and that T lies in the interval (t, t + δ) for
arbitrarily small δ.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.