+1-415-315-9853

info@mywordsolution.com

## Statistics

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

(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

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?

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

(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.

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.

Statistics and Probability, Statistics

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

Have any Question?

## Related Questions in Statistics and Probability

### Develop a research question that addresses one of the

Develop a research question that addresses one of the unknowns in the question: Positive interactions between the CEO and board members lead to enhanced and better performance of the Corporation . Address the following 1 ...

### In a poll of 2003 adults within the united states for the

In a poll of 2,003 adults within the United States for the Wall Street Journal, 57% of adults said that public education needs to be improved. Construct a 95% confidence interval for the proportion of all U. S. adults wh ...

### A random sample of the price of gasoline from 40 gas

A random sample of the price of gasoline from 40 gas stations in a region gives the statistics below. Complete parts a) and b). ybar = \$3.79, s = \$0.27 a) Find a 95% confidence interval for the mean price of regular gaso ...

### The condorcet method the french mathematician condorcet

The Condorcet Method The French mathematician Condorcet proposed the following method for determining a social preference order based on the strict preferences of the individuals in society. A voter i with a strict prefe ...

### The mean number of medicine taken by a random sample of 26

The mean number of medicine taken by a random sample of 26 patients has a standard deviation of 3.1 medications. Assume the population is normally distributed Construct a 99% confidence interval for the population standa ...

### Hello kitty company claims that the proportion of its

Hello Kitty Company claims that the proportion of its employees investing in individual investment accounts is higher than national proportion of 50%. A survey of 100 employees in Hello Kitty Company indicated that 58 of ...

### Business quantative analysis and statistics 1project

BUSINESS QUANTATIVE ANALYSIS AND STATISTICS 1 Project Management Scenario: You are a consultant with expertise in project management strategies. Select a possible project management situation from your current or future ...

### The mean incubation time for a type of fertilized egg kept

The mean incubation time for a type of fertilized egg kept at 100.7° F is 22days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 2 days. (a) The probability that a r ...

### You are at a magic show the magician has a coin and you as

You are at a magic show. The magician has a coin and you, as a member of the audience, do not know whether the coin has heads on both sides or heads on one side and tails on the other. The null hypothesis is that the coi ...

### Consider rolling a fair 6-faced die twice let a be the

Consider rolling a fair 6-faced die twice. Let A be the event that the sum of the two rolls is atmost 6, and B be the event that the first one is an even number. (a) What is the probability that the sum of the two rolls ...

• 13,132 Experts

## 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.

### 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