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 pij(t) = P(X(t) = j|X(0) = 1). Explain why p12(t) = P13(t) = P14(t) = P15(t) (nocomputations required).c. Write the forward Chapman-Kolmogorov equation, and prove thatPin(0) =154-Pu (1).

Given Information: Consider a given study in which a frog in a pond with 5 water lilies numbered…

Answered: A frog is in a pond with 5 water lilies…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.5: Markov Chain

Problem 49E: Consider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show...

See similar textbooks

Similar questions

Consider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show that the steady state matrix X depends on the initial state matrix X0 by finding X for each X0. X0=[0.250.250.250.25] b X0=[0.250.250.400.10] Example 7 Finding Steady State Matrices of Absorbing Markov Chains Find the steady state matrix X of each absorbing Markov chain with matrix of transition probabilities P. b.P=[0.500.200.210.300.100.400.200.11]
Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).
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)
If Kt = B2t - t, where B is standard Brownian Motion, show that Kt is a martingale, and a markov process
For the attached transition probability matrix for a Markov chain with {Xn ; n = 0, 1, 2,.........}: a) How many classes exist, and which two states are the absorption states? b) What is the limn->inf P{Xn = 3 | X0 = 3}? b) What is the limn->inf P{Xn = 1 | X0 = 3}?
Let {X(t);t≥0} be a Continuous Time Markov chain with state space S={1,2} . The process is such that after an exponential random time with parameter λ=1 , it moves from state 1 to state 2, and then remains at state 2. (a) The entries of the generator matrix Q are q11 Answer q12 Answer q21 Answer q22 Answer (b) Based on results for part (a); the transition probability matrix P(3) (i.e at time t=3 ) is (round off probabilty to two decimal points) p11 Answer p12 Answer p21 Answer p22 Answer
Suppose that X k is a time-homogenous Markov chain. Show thatP{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3.(a) Find P(X2 = 1).(b) Find the conditional probability P(X2 = 1|X1 = 1).(c) Find the conditional probability P(X1 = 1|X2 = 1).(d) Find limn→∞ P(Xn = 1).

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS