1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by -4 1 Q = 0 3 0 (a) Complete the matrix Q. Draw the transition diagram of X. (b) Find the average time X spends in each state and the probability of making a jump between states. (c) Let q₁ = -qii. Use the following integrated form of Kolmogorov backward equation to find the transition (probability) matrix [P,]ij = P(X = j\X₁ = i) of X, Pij(t) = Pij(0)eit + k#i ΣΕ -qu e "qik Pkj(t - u)du. (d) Verify your solution for the transition probability pij(t) by checking the limit Q = lim where I is a (3 × 3)-identity matrix. (P,-I) 10 t
1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by -4 1 Q = 0 3 0 (a) Complete the matrix Q. Draw the transition diagram of X. (b) Find the average time X spends in each state and the probability of making a jump between states. (c) Let q₁ = -qii. Use the following integrated form of Kolmogorov backward equation to find the transition (probability) matrix [P,]ij = P(X = j\X₁ = i) of X, Pij(t) = Pij(0)eit + k#i ΣΕ -qu e "qik Pkj(t - u)du. (d) Verify your solution for the transition probability pij(t) by checking the limit Q = lim where I is a (3 × 3)-identity matrix. (P,-I) 10 t
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 14EQ
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![1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
-4
1
Q =
0
3
0
(a) Complete the matrix Q. Draw the transition diagram of X.
(b) Find the average time X spends in each state and the probability of making a jump
between states.
(c) Let q₁ = -qii. Use the following integrated form of Kolmogorov backward equation
to find the transition (probability) matrix [P,]ij = P(X = j\X₁ = i) of X,
Pij(t) = Pij(0)eit
+
k#i
ΣΕ
-qu
e "qik Pkj(t - u)du.
(d) Verify your solution for the transition probability pij(t) by checking the limit
Q = lim
where I is a (3 × 3)-identity matrix.
(P,-I)
10 t](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F222cca7b-b117-4d0c-a3e7-fdec654c3a21%2F8tdtc0j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
-4
1
Q =
0
3
0
(a) Complete the matrix Q. Draw the transition diagram of X.
(b) Find the average time X spends in each state and the probability of making a jump
between states.
(c) Let q₁ = -qii. Use the following integrated form of Kolmogorov backward equation
to find the transition (probability) matrix [P,]ij = P(X = j\X₁ = i) of X,
Pij(t) = Pij(0)eit
+
k#i
ΣΕ
-qu
e "qik Pkj(t - u)du.
(d) Verify your solution for the transition probability pij(t) by checking the limit
Q = lim
where I is a (3 × 3)-identity matrix.
(P,-I)
10 t
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