4. Consider a continuous-time Markov chain X(t) on S = {1,2,3} with associated em-bedded chain given by0 1 0P = |0 0 11.22Assume the holding time parameters are given by A1 = 2, A2 = 1, A3 = 3.(a) Find the limiting distribution u of the embedded chain.(b) Using u, find the stationary distribution for X(t).(c) Verify your answer in part (b) by finding the generator matrix for this chain andthen solve TQ = 0

Answered: Image /qna-images/answer/6540129c-33d9-4105-943f-0ff1f9fc00cc.jpg

Answered: Consider a continuous-time Markov chain…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

2.2 Let Xo. X1,... be a Markov chain with transition matrix 1 2 3 1 1/2 1/2) 1 3(1/3 1/3 1/3) and initial distribution a = (1/2,0, 1/2). Find the following: (a) P(X, = 1|X¡ = 3) (b) P(X1 = 3, X2 = 1) (c) P(X1 = 3|X2 = 1) (d) P(X9 = 1|X1 = 3, X4 = 1,X7 = 2) %3D
3.4 Consider a Markov chain with transition matrix 1- a a P = 1-b b 1-c where 0< a, b, c < 1. Find the stationary distribution.
A random process is defined by x(t) = A, where A is a continuous random variable uniformly distributed on (0, 1). (1) Determine the form of the sample function (ii) Classify the process (iii) Is it deterministic?
3. Let X ~ Exponential() and let t be a constant with 0 0 be any value. (a) Calculate P(X > b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= , use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etx). (d) Next, use the Markov inequality to prove a bound on P(etX > a) (here a > 0 is any positive number, while we assume 0 b) (here b > 0 is any positive number, and again 0 b) as in part (a), the upper bound on P(X > b) as in part (b), and the alternative upper bound on P(X > b) as in part (e). How close are these upper bounds to the real probability? Which bound is better? Try this for a few values of X, b,t and describe what you see. 6.
Suppose that a Markov chain has transition probability matrix 1 2 1 P (1/2 1/2 2 1/4 3/4 (a) What is the long-run proportion of time that the chain is in state i, i = 1,2 ? 5. What should r2 be if it is desired to have the long-run average (b) Suppose that ri reward per unit time equal to 9?
2. Let Xo, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix (1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6/ (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it.
Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
Let X, X2,..., Xg be i.i.d random variables from a distribution with p.d.f co"e , z>0 otherwise, f(r) = where c is a constant that makes this a valid p.d.f.. Let Y=YX². Use Markov's inequality to find an upper bound for P(Y > 1260).
Let X, be the Markov chain with state space Z and transition probability Pa,z+1 = P, Pa,2-1 = 1- p, where p > 1/2. Assume X, = 0. (a) Let Y = min{Xo, X1,...}. What is the distribution of Y? (b) For positive integer k, let T = min{n : X, = k} and let e(k) = E[TR]. Explain why e(k) = ke(1). (c) Find e(1). Hint: part (b) might be helpful. (d) Use (c) to give another proof that e(1) = o if p = 1/2.
2. Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 2/3 1/6 1/6, 0 0 (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems in the notes. (b) Find the limiting distribution for this Markov chain. (c) Without doing any more calculations, what can you say about p1100) and p(100)?
Consider a (non)-symmetric random walk M = Xo +X1+X2+…+Xn where X;'s (i> 1) are independent and identical distributed, P(X; = 1) = p and P(X;=-1) =1–p and Xo = 0. Let Sn = eaS,-n Denote Fn= 0(X1,...,Xn). Find a such that (Sn)n20 is a (Fn)n20 - martingale.
Suppose that a Markov chain (X,)n>o has a stochastic matrix given by: 1/2 1/2 1/4 3/4 1/3 1/3 1/3 P = 1/6 2/3 1/6 2/3 1/3 1/6 5/6 1/2 0 0 0 1/2 (c) What is P4(H' < H')? (d) What is P2(H' < H')? (e) Suppose now that the Markov chain is initialised from 1, which is a uniform distribution over all states 1 to 7. What is Pa(H' < H')? (f) What is the probability that a Markov chain, which started in state 2, visits state 1 exactly once before hitting state 7?

SEE MORE QUESTIONS

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS