10. Define the mean value of a discrete random variable and the probability generating function . Show that the mean value is (1). If (s)

10. Define the mean value of a discrete random variable and the probability generating function φ.

Show that the mean value is φ′(1). If φ(s) has the form p(s)/q(s) show that the mean value is

(p′(1) − q′(1))/q(1).

Two duellists, A and B, fire at each other in turn until one hits the other. Each duellist has the same probability of obtaining a hit with each shot fired, these probabilities being a for A and b for B. If A fires the first shot, calculate the probability that A wins the duel. Find also the probability distribution of the number of shots fired before the duel terminates. What is the expected number of shots fired? (Oxford 1976M)

60 Probability generating functions