5 Consider the success run chain in Example 8.2.16. Suppose that the chain has been

running for a while and is currently in state 10. (a) What is the expected number of

steps until the chain is back at state 10? (b) What is the expected number of times the

chain visits state 9 before it is back at 10?

16 Consider a version of the success run chain in Example 8.2.16 where we disregard

sequences of consecutive tails, in the sense that for example T, T T, T T T, and so on,

all simply count as T. Describe this as a Markov chain and examine it in terms of

irreducibility, recurrence, and periodicity. Find the stationary distribution and compare

with Example 8.2.16. Is it the limit distribution?

17 Reversibility. Consider an ergodic Markov chain, observed at a late timepoint n. If

we look at the chain backward, we have the backward transition probability qij =

P(Xn?1 = j|Xn = i). (a) Express qij in terms of the forward transition probabilities

and the stationary distribution ?. (b)If the forward and backward transition probabilities

are equal, the chain is called reversible. Show that this occurs if and only if ?ipij =

?jpji for all states i, j (this identity is usually taken as the definition of reversibility).

(c) Show that if a probability distribution ? satisfies the equation ?ipij = ?jpji for all

i, j, then ? is stationary.

18 The intuition behind reversibility is that if we are given a sequence of consecutive states

under stationary conditions, there is no way to decide whether the states are given in

forward or backward time. Consider the ON/OFF system in Example 8.2.4; use the

definition in the previous problem to show that it is reversible and explain intuitively.

19 For which values of p is the following matrix the transition matrix of a reversible Markov

chain? Explain intuitively.

P =

0 p 1 ? p

1 ? p 0 p

p 1 ? p 0

!

20 Ehrenfest model of diffusion. Consider two containers containing a total of N gas

molecules, connected by a narrow aperture. Each time unit, one of the N molecules is

chosen at random to pass through the aperture from one container to the other. Let Xn

be the number of molecules in the first container. (a) Find the transition probabilities

for the Markov chain {Xn}. (b) Argue intuitively why the chain is reversible and why

the stationary distribution is a certain binomial distribution. Then use Problem 17 to

show that it is indeed the stationary distribution. (c) Is the stationary distribution also

the limit distribution?

21 Consider an irreducible and positive recurrent Markov chain with stationary distribution

? and let g : S ? R be a real-valued function on the state space. It can be shown that

1

n

Xn

k=1

g(Xk)

P?

X

j?S

g(j)?j

for any initial distribution, where we recall convergence in probability from Section 4.2.

This result is reminiscent of the law of large numbers, but the summands are not i.i.d.

We have mentioned that the interpretation of the stationary distribution is the long-term

proportion of time spent in each state. Show how a particular choice of the function g

above gives this interpretation (note that we do not assume aperiodicity)

2. Suppose that Y, and Yz are continuous random variables with the joint probability density function (joint pdf) f(1. Y2) = ky; (y, + 2):0 0. 4. Use the Fundamental Theorem to determine the value of b if the area under the graph of f(x) = 4x between x = 1 and x = b is equal to 240. Assume b > 1. 2005. worldwideThe following table shows the average hourly wage rates for day-care centers from two locations based on two random samples. Use the table to complete parts a through c. Location 1 Location 2 Sample mean $9.62 $8.59 Sample standard deviation $1.23 $1.11 Sample size 28 35 a. Perform a hypothesis test, using a =0.10, to determine if the average hourly wage for day-care workers in Location 1 is $0.50 per hour higher than the average hourly wage for day-care workers in Location 2. Assume the population variances for wage rates in each location are equal. Determine the null and alternative hypotheses for the test. Ho: 1 - H2 5 $0.50 H1: 1 - 12 > $0.50 Calculate the appropriate lost statistic and interpret the result. The test statistic is (Round to two decimal places as needed.) Enter your answer in the answer box and then click Check Answer.A researcher wishes to determine whether listening to classical music improves students' performance on a memory test. Forty-six students are randomly selected to perform a memory test once while listening to classical music and once without listening to music. Test at the 95% confidence level. Identify the null and alternative hypotheses, critical value, test statistic, P-value, the decision regarding the hypothesis, and the final conclusion. Draw the distribution. Provide the Confidence Interval and interpret. You are provided with summary information: the mean difference in the sample scores was 3.7 with a standard deviation of the sample difference of 4.4