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A generator for a continuous time Markov process X(t) is given by G = 2 2 ー人 (1 0 a

A generator for a continuous time Markov process X(t) is given by G = 2 2 ー人 (1 0 a

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 49E: Consider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show...
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A generator for a continuous time Markov process X(t) is given by G The states are {1, 2, 3}, so e.g. g12 = A and g23 = A, etc. (i) Write down the probabilities of moving from one state to another state after the length of stay in a particular state is complete. (ii) Show that P(X(t + h) = 3|X(t) = 1) lim h→0 as h → 0. h (iii) The stationary probability vector T satisfies T P(t) P(t) is the matrix of probabilities given by pij(t) = P(X(t) = j|X (0) = i). Find 7 and show that a G = 0. = T for all t, where (iv) At time t = 2 the process is in state 1; how much longer does it stay in this state. (v) Given X(0) = 1, find the probability that the process has not visited state 3 by time t = 4.
Transcribed Image Text:A generator for a continuous time Markov process X(t) is given by G The states are {1, 2, 3}, so e.g. g12 = A and g23 = A, etc. (i) Write down the probabilities of moving from one state to another state after the length of stay in a particular state is complete. (ii) Show that P(X(t + h) = 3|X(t) = 1) lim h→0 as h → 0. h (iii) The stationary probability vector T satisfies T P(t) P(t) is the matrix of probabilities given by pij(t) = P(X(t) = j|X (0) = i). Find 7 and show that a G = 0. = T for all t, where (iv) At time t = 2 the process is in state 1; how much longer does it stay in this state. (v) Given X(0) = 1, find the probability that the process has not visited state 3 by time t = 4.
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