Random incidence in the Bernoulli process. Your cousin has

QUESTION:

Random incidence in the Bernoulli process. Your cousin has been playing the same video game from time immemorial. Assume that he wins each game with probability p. independent of the outcomes of other games. At midnight, you enter his room and witness his losing the current game. What is the PMF of the number of lost games between his most recent win and his first future win? Solution. Let t be the numher of the game when you enter the room. Let !vI be the numher of the most recent pa.. c;t game that he won, and let N be the number of the first game to be won in the future. The random variable X = N - t is geometrically distributed with parameter p. By symmetry and independence of the games, the random variahle Y = t - M is also geometrically distributed with parameter p. The games he lost hetween his most recent win and his first future win are all the games between M and N. Their number L is given by L = N - Al - 1 = X + Y - 1. Thus. L + 1 has a Pascal PMF of order two, and k = 2,3, ... . Hence. pL(i) = P(L + 1 = i + 1) = i p2 (1 _ p)' - I. i = 1,2 .... .