For each of the following statements, enter T if the statement is true, enter F is the statement is false. DO NOT ENTER "True" or "False". (a) If a branching process (Zn) with Zo = 1 has offspring distribution pgf G(s) satisfying G' (1) = 1 then the extinction probability must be 1. (b) If a branching process (Zn) with Zo = 1 has offspring distribution pgf G(s) satisfying G(0) = 0 then the extinction probability must be 1. (c) Let C and C be two distinct communicating classes of a Markov chain, and suppose there are two particular states a and b such that a is in C and b is in C and a b. Then for all states i in C and j in C, i j. (d) If a finite state space, irreducible Markov chain has a unique stationary distribution, then it has a limiting distribution which is equal to this stationary distribution. (e) A Markov chain with a finite state space must have at least one recurrent state. (f) If i, j are states in a Markov chain, i is recurrent and i leads to j, then j must lead to i. (g) If S and S3 are the second and third arrival times in a rate Poisson process then E(SS3) = E(S)E(S3). (g) If X is an exponential random variable with parameter 1/2, then Pr(X > 2h) = = o(h) as h0. (i) If (N) is a rate & Poisson process, and 0 < s < t, then Pr(N = 3, N = 4) = Pr(N = 3, Nt-s = 1). (i) If (N) is a rate > Poisson process, and 0 < s < t, then Pr(N = 3, N = 4) = Pr(N = 3)Pr(Nt-s (k) The sum of two independent Poisson processes is a Poisson process. = 1).