For each of the following statements, enter T if the statement is true, enter F is...

 

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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 C₁ and C₂ be two distinct communicating classes of a Markov chain, and suppose there are two particular states a and b such that a is in C₁ and b is in C₂ and a → b. Then for all states i in C₁ and j in C₂, 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 S₂ and S3 are the second and third arrival times in a rate Poisson process then E(S₂S3) = E(S₂)E(S3). (g) If X is an exponential random variable with parameter 1/2, then Pr(X > 2h) = = o(h) as h→0. (i) If (N₂) is a rate & Poisson process, and 0 < s < t, then Pr(N = 3, N₁ = 4) = Pr(N₁ = 3, Nt-s = 1). (i) If (N₂) is a rate > Poisson process, and 0 < s < t, then Pr(N = 3, N₁ = 4) = Pr(N = 3)Pr(Nt-s (k) The sum of two independent Poisson processes is a Poisson process. = 1). 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 C₁ and C₂ be two distinct communicating classes of a Markov chain, and suppose there are two particular states a and b such that a is in C₁ and b is in C₂ and a → b. Then for all states i in C₁ and j in C₂, 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 S₂ and S3 are the second and third arrival times in a rate Poisson process then E(S₂S3) = E(S₂)E(S3). (g) If X is an exponential random variable with parameter 1/2, then Pr(X > 2h) = = o(h) as h→0. (i) If (N₂) is a rate & Poisson process, and 0 < s < t, then Pr(N = 3, N₁ = 4) = Pr(N₁ = 3, Nt-s = 1). (i) If (N₂) is a rate > Poisson process, and 0 < s < t, then Pr(N = 3, N₁ = 4) = Pr(N = 3)Pr(Nt-s (k) The sum of two independent Poisson processes is a Poisson process. = 1).

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a If a branching process Z with Zo 1 has offspring distribution pgf Gs satisfying G1 1 then the extinction probability must be 1 Explanation This statement is false The condition G1 1 is necessary for ... View the full answer