Consider a population of N individuals, through which an epidemic passes until all individuals have become infected. Suppose that at time 0 one member of the population is infected and the remaining individuals are susceptible. The only changes possible are when a susceptible individual becomes infected. Once infected, an individual remains infected, and infectious, thereafter. Ftom the time when an individual becomes infected, the time until he or she infects each susceptible individual is an exponentially distributed random variable (with parameter one, for simplicity), and the infection times of different pairs of individuals are independent of each other. Let the duration of the epidemic be denoted by T=U1+U2+U3+..-+UN1, where Uh, is the amount of time spent with exactly 2' individuals infected. [a] What is the distribution of U1; for each possible value of k? [1)] Calculate E[T]. -n- 1|- [c] Does the expected duration of the epidemic increase or decrease as a function of N '2'