1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by -4 ... Q 0 ... 1 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 = q. Use the following integrated form of Kolmogorov backward equation qi to find the transition (probability) matrix [P;];; = P(X₁ = j|X₁ = i) of X, Pij(t) = Pij(0)e-gi² + Σ√ √ eqikPkj(t-u)du. k#i (d) Verify your solution for the transition probability pi;(t) by checking the limit (Pt-I) Q = lim to t where I is a (3 × 3)-identity matrix. (1)
1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by -4 ... Q 0 ... 1 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 = q. Use the following integrated form of Kolmogorov backward equation qi to find the transition (probability) matrix [P;];; = P(X₁ = j|X₁ = i) of X, Pij(t) = Pij(0)e-gi² + Σ√ √ eqikPkj(t-u)du. k#i (d) Verify your solution for the transition probability pi;(t) by checking the limit (Pt-I) Q = lim to t where I is a (3 × 3)-identity matrix. (1)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
-4
...
Q
0
...
1
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 = q. Use the following integrated form of Kolmogorov backward equation
qi
to find the transition (probability) matrix [P;];; = P(X₁ = j|X₁ = i) of X,
Pij(t) = Pij(0)e-gi² +
Σ√ √
eqikPkj(t-u)du.
k#i
(d) Verify your solution for the transition probability pi;(t) by checking the limit
(Pt-I)
Q = lim
to
t
where I is a (3 × 3)-identity matrix.
(1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ecbf7c4-5c4a-42d0-9733-4e95edbd9757%2F89a6b883-7249-4613-afab-335b9528da47%2Fp35uc4l_processed.png&w=3840&q=75)
Transcribed Image Text:1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
-4
...
Q
0
...
1
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 = q. Use the following integrated form of Kolmogorov backward equation
qi
to find the transition (probability) matrix [P;];; = P(X₁ = j|X₁ = i) of X,
Pij(t) = Pij(0)e-gi² +
Σ√ √
eqikPkj(t-u)du.
k#i
(d) Verify your solution for the transition probability pi;(t) by checking the limit
(Pt-I)
Q = lim
to
t
where I is a (3 × 3)-identity matrix.
(1)
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