1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by 여 -4 Q = 0 ... 1 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 = −qi. Use the following integrated form of Kolmogorov backward equation to find the transition (probability) matrix [Pr]ij = P(X₁ = j\Xo = i) of X, Pij(t) = Pij(0)e¯qit + Σ e-giu 'Qik Pkj(t - u)du. k#i 0 (d) Verify your solution for the transition probability pij(t) by checking the limit (P₁-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 여 -4 Q = 0 ... 1 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 = −qi. Use the following integrated form of Kolmogorov backward equation to find the transition (probability) matrix [Pr]ij = P(X₁ = j\Xo = i) of X, Pij(t) = Pij(0)e¯qit + Σ e-giu 'Qik Pkj(t - u)du. k#i 0 (d) Verify your solution for the transition probability pij(t) by checking the limit (P₁-I) Q = lim , to t where I is a (3 × 3)-identity matrix. (1)
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Please do question 1b, 1c and 1d with full working out. I'm struggling to understand what to write
![1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
여
-4
Q = 0
...
1
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 = −qi. Use the following integrated form of Kolmogorov backward equation
to find the transition (probability) matrix [Pr]ij = P(X₁ = j\Xo = i) of X,
Pij(t) = Pij(0)e¯qit +
Σ
e-giu
'Qik Pkj(t - u)du.
k#i
0
(d) Verify your solution for the transition probability pij(t) by checking the limit
(P₁-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%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F4d16c6f9-2e11-499d-a881-f2ccc1e7a192%2Fnm00kds_processed.png&w=3840&q=75)
Transcribed Image Text:1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by
여
-4
Q = 0
...
1
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 = −qi. Use the following integrated form of Kolmogorov backward equation
to find the transition (probability) matrix [Pr]ij = P(X₁ = j\Xo = i) of X,
Pij(t) = Pij(0)e¯qit +
Σ
e-giu
'Qik Pkj(t - u)du.
k#i
0
(d) Verify your solution for the transition probability pij(t) by checking the limit
(P₁-I)
Q
= lim
,
to
t
where I is a (3 × 3)-identity matrix.
(1)
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