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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 15EQ

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Problem 2
A frog is in a pond with 5 water lilies numbered from 1 to 5. With exponential rate 1, the frog
leaves its current water lily, chooses a new one uniformly among the four others and jumps to it.
We assume the frog starts from lily 1. Let X(t) be the number of the lily where the frog is at time
t.
a. Admitting that X(t) is a continuous-time Markov chain, give its parameters (i.e. the vi and Pij
of the course). No proof is required.
b. Let pij(t) = P (X(t) = j|X(0) = 1). Explain why p12(t)
P13(t) = P14(t) = P15(t) (no
computations required).
c. Write the forward
d.
Chapman-Kolmogorov equation, and prove
5
-Pu(t).
Pir(t) =
1
=
4
Solve this equation to compute p₁1(t).
that

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Question 3. Recall that the discrete stochastic process {Xn, n = 0, 1, 2,...} is a Markov chain if for each n, P(Xn+1 = j|Xn = i, Xn-1 = in-1,..., Xo = io) = P(Xn+1 = j|Xn = i) = Pij. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following hold as well? P(Xn+2=jXn = i, Xn-1 = in-1, ..., Xo = io) = P(Xn+2 = j|Xn = i) Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof. Please step by step answer.
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A frog is in a pond with 5 water lilies numbered from 1 to 5. With exponential rate 1, the frogleaves its current water lily, chooses a new one uniformly among the four others and jumps to it.We assume the frog starts from lily 1. Let X(t) be the number of the lily where the frog is at timet.a. Admitting that X(t) is a continuous-time Markov chain, give its parameters (i.e. the vi and pijof the course). No proof is required.b. Let p1j (t) = P (X(t) = j|X(0) = 1). Explain why p12(t) = p13(t) = p14(t) = p15(t) (nocomputations required).c. Write the forwards Chapman–Kolmogorov equation, and prove thatp′11(t) = 14 − 54 p11(t).d. Solve this equation to compute p11(t)
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