A generator for a continuous time Markov process X(t) is given by G = 2 2 ー人 (1 0 a
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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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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