Duration : 120 Minutes Answer all questions. 1. Consider a Poisson process {N (t) : t 2 0} with intensity )v. > 0. Recall that the density of such a. Poisson process is til pr{Nit} =J'}= (2,) e'\". j=0,1,2,... Let the time of the nth arrival be dened by '1'".I = inf{t : N(i) = n}, and the nth inter-arrival time [i.e. duration) be X\" = Tn T..-1. (a) State and interpret the three conditions that dene a Poisson prom. (1)] Draw a picture to illustrate the relations between NU). T... and X... (c) Show that the Poisson process; is a Markov process. Find the transition probabilities maa) and Pepsi\"- (d) Consider Pr{X1 > t), and hence obtain the distribution of X1. Consider Pr(X2 > i|X1 = t1), and hence obtain the distribution of X3. (e) Find P1111