A runaway chain reaction can occur when the per-capita activation rate, i.e., the rate at which each active particle activates other particles, increases quickly enough with respect to the number of active particles. Let's model this to understand it a bit better. a) Let rk denote the per-capita activation rate when there are k active particles. Based on the de- scription above, explain why, for k 1, the waiting time between activation of the kth and k+ 1st particles is exponential with rate k rk. b) Suppose that the per-capita activation rate is equal to the number of active particles. Let Tk be the activation time of the kth particle, with T1 = 0, and let tk = Tk Tk1. Give the distribution of the sequence (tk) (you can assume the (tk) are independent of each other). c) We can define the explosion time by T := limkTk. With (rk) as in part (b), is the expected explosion time finite or infinite?