Ask Question, Ask an Expert

+1-415-315-9853

info@mywordsolution.com

Ask Statistics and Probability Expert

1) Consider a Markov chain C1,C2, . . . .

(a) Prove that the joint probability of the first n steps is

p(C1, . . . ,Cn) = p(C1)πt=2p(Ct| Ct-1).

(b) Hence prove that, for any r < s < t p(Cr, Cs,Ct) = p(Cr) p(Cs|Cr) p(Ct | Cs).

(c) Hence prove that Cr and Ct are conditionally independent given Cs.

2) For an m-state Markov chain Ct with transition matrix G,
(a) prove that the marginal state probabilities at times t and t + 1 are related through

u(t + 1) = u(t) τ,

where u(t) is row vector (p(Ct = 1), . . . , p(Ct = m)).

(b) Hence prove that u(t + k) = u(t) τk                          (*)

for any integer k>=0.

(c) Under what conditions would (*) hold for k < 0?

3) For a general discrete homogenous Markov chain with transition matrix G and a specified initial distribution u(1), prepare an R function to do the following:

(a) compute the marginal distribution u(t) from t = 1 up to a specified time t = n

(b) plot the results.

• use operator %*% for matrix multiplication

• use rbind to collect vectors into rows of a matrix U

• use matplot(c(1:n),U,type="l") to plot them in a single plot.

Your R function should contain plenty of comments documenting its inputs, outputs and internal steps.

4) A discrete homogenous Markov chain has transition matrix

1178_Transition matrix.jpg

(a) Using your R function from Exercise 4, plot the marginal distribution u(t) for t = 1, . . . , 20, assuming the chain starts in state 1.

(b) Repeat part (a), assuming the chain starts in state 4.

(c) Deduce theoretically the stationary distribution δ of this chain.

(d) Compare your results from parts (a)–(c).

5) A discrete homogenous Markov chain has transition matrix

587_Transition matrix_1.jpg

where 0 < a < 1.

(a) Show that  δ= ( 1/3 , 1/3 , 1/3) is a stationary distribution of this Markov chain.

(b) Are there other stationary distributions of this chain?

(c) Comment on your conclusion.

6)a) prepare an R function to generate a sequence of a given length from an m-state homogenous Markov chain with given transition matrix and initial state. As always, your code should contain plenty of comments. Hint: use sample(c(1:m), size=1, prob=p) to sample a single value from distribution p.

(b) Use this function to generate a sequence of length 100 from a Markov chain with initial state 1 and transition matrix

685_Transition matrix_2.jpg

(c) From the resulting sequence, estimate the transition matrix, showing all working.

(d) Compare the true and estimated transition matrices.

7) (a) prepare an R function to estimate the transition matrix of a discrete homogenous Markov chain from a given sequence of observations.

(b) Repeat Exercise E2.7b, this time generating a sequence of length 1000. For each value of k in 100, 200, . . . , 1000, estimate the transition matrix from only the first k observations, using your function from part (a).

(c) Plot these estimated transition matrices against k. Hint: use function matplot.

(d) Comment on your results.

8) N independent sequences, each of length n, are generated from a discrete homogenous Markov chain with transition matrix τ, with initial states drawn from an unknown distribution f .

(a) prepare down the likelihood for such data.

(b) Derive the MLEs of f and τ.

(c) Comment on your results.

Statistics and Probability, Statistics

  • Category:- Statistics and Probability
  • Reference No.:- M91861

Have any Question? 


Related Questions in Statistics and Probability

The mixed methods approachthrough your work in this unit

The Mixed Methods Approach Through your work in this unit, you have come to understand just how different quantitative and qualitative decision-making techniques are. Perhaps that is why a mixed methods decision-making t ...

Marjorie bought a home with an adjustable-rate mortgage the

Marjorie bought a home with an adjustable-rate mortgage. The margin on an adjustable-rate mortgage is 2.5% and the rate cap is 7% over the life of the loan. If the current index rate is 5.9%, what is the calculated inter ...

Assignment- inferences and linear regression1a left-tail

Assignment- Inferences and Linear Regression 1. A left-tail area in the chi-square distribution equals 0.95. For df = 10, the table value equals A. 18.307. B. 20.483. C. 3.940. D. 15.987. 2. Which of the following statem ...

A spinoff of the anti-war and civil rights movements of the

A spinoff of the anti-war and civil rights movements of the 1960s that affected women was? increased desire to return to the security of homes and families decreased political involvement. the birth of the women's libera ...

How did religion shape indian reaction to white settlement

How did religion shape Indian reaction to white settlement in the nineteenth century?

Suppose that the number of men equals the number of women

Suppose that the number of men equals the number of women. Prove the following claims, or provide counterexamples: (a) For every pair of matchings there exist preference relations for which these are two stable matchings ...

Explain how you would conduct the study you designed in

Explain how you would conduct the study you designed in Question as a matched-subjects design. Question:- Design a randomized groups experiment to test the hypothesis that children who watch an hour-long violent televisi ...

When conducting a hypothesis test if z calculated is less

When conducting a hypothesis test, if z calculated is less than z critical, you should accept the null hypothesis?

Assignment choice 1 the great migrationwrite an essay on

Assignment Choice #1: The Great Migration Write an essay on the great migration of the black population to the northern United States.  What were the reasons for the great migration? What were some of the risks involved ...

A parcel of land purchased by clean energy inc is estimated

A parcel of land purchased by Clean Energy, Inc. is estimated to contain 15.25 million tons of anthracite coal. The purchase price was $10,850,000, startup costs were $3,411,000, and the residual value is $1,464,000. If ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

WalMart Identification of theory and critical discussion

Drawing on the prescribed text and/or relevant academic literature, produce a paper which discusses the nature of group

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro