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1. Let Z = (Z₁: = 0, 1..) be a Markov chain defined on a state space S = (1. m). Let the one-step transition probability matrix M of Z have zero diagonal element, i.e., M(i, j)= Jo, j =i for any state (i, j) € S with qij 20 and qi = Σ9 > 0. Let N = (N₁:12 0} be a time-homogeneous Poisson process with intensity >0 and No = 0. Assume that Z and N are independent. Consider a stochastic process X₁ = ZN, for 120. (1) The (i. j)-element of transition matrix P(r) of X is defined by the conditional probability P(X, = j|Xo = i) = P(Z = j, N₁ = n|Zo = i). m=0 (a) Show that X is a continuous-time Markov chain with transition matrix P(t) = e where the matrix Q is defined by Q = D(M-I), where I is a (m x m)-identity matrix and D is a diagonal matrix diag(q₁qm). Hint for a square matrix A, E = e¹. (b) Show that is an intensity matrix. (2)

1. Let Z = (Z₁: = 0, 1..) be a Markov chain defined on a state space S = (1. m). Let the one-step transition probability matrix M of Z have zero diagonal element, i.e., M(i, j)= Jo, j =i for any state (i, j) € S with qij 20 and qi = Σ9 > 0. Let N = (N₁:12 0} be a time-homogeneous Poisson process with intensity >0 and No = 0. Assume that Z and N are independent. Consider a stochastic process X₁ = ZN, for 120. (1) The (i. j)-element of transition matrix P(r) of X is defined by the conditional probability P(X, = j|Xo = i) = P(Z = j, N₁ = n|Zo = i). m=0 (a) Show that X is a continuous-time Markov chain with transition matrix P(t) = e where the matrix Q is defined by Q = D(M-I), where I is a (m x m)-identity matrix and D is a diagonal matrix diag(q₁qm). Hint for a square matrix A, E = e¹. (b) Show that is an intensity matrix. (2)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1. Let Z = = {Zn: n = 0, 1, } be a Markov chain defined on a state space S {1, ,m}. Let the one-step transition probability matrix M of Z have zero diagonal element, i.e., M(i, j) = 0, qij qi j=i j‡i, = for any state (i, j) € S with qij ≥ 0 and qi Σjti qij > 0. Let N = {N₁ t≥ 0} be a time-homogeneous Poisson process with intensity > 0 and No = 0. Assume that Z and N are independent. Consider a stochastic process X₁ZN₁, for t≥ 0. = (1) The (i, j)-element of transition matrix P(t) of X is defined by the conditional probability where the matrix Q is defined by CO P(X, = j|Xo = i) = Σ P(Zn = j, N, = n|Zo = i). n=0 (a) Show that X is a continuous-time Markov chain with transition matrix P(t) = eºt, (2) Q = D(M-I), where I is a (m x m)-identity matrix and D is a diagonal matrix diag(q₁,..., 9m). Hint for a square matrix A, E = ¹. A" n! (b) Show that Q is an intensity matrix.
Transcribed Image Text:1. Let Z = = {Zn: n = 0, 1, } be a Markov chain defined on a state space S {1, ,m}. Let the one-step transition probability matrix M of Z have zero diagonal element, i.e., M(i, j) = 0, qij qi j=i j‡i, = for any state (i, j) € S with qij ≥ 0 and qi Σjti qij > 0. Let N = {N₁ t≥ 0} be a time-homogeneous Poisson process with intensity > 0 and No = 0. Assume that Z and N are independent. Consider a stochastic process X₁ZN₁, for t≥ 0. = (1) The (i, j)-element of transition matrix P(t) of X is defined by the conditional probability where the matrix Q is defined by CO P(X, = j|Xo = i) = Σ P(Zn = j, N, = n|Zo = i). n=0 (a) Show that X is a continuous-time Markov chain with transition matrix P(t) = eºt, (2) Q = D(M-I), where I is a (m x m)-identity matrix and D is a diagonal matrix diag(q₁,..., 9m). Hint for a square matrix A, E = ¹. A" n! (b) Show that Q is an intensity matrix.
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