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)

What is health financing? What are its desirable and undesirable effects? Discuss how cost-sharing applies to health insurance. What is Medicare Part A? Discuss the financing Medicare Part A. What services does Part A cover? Discuss the payment method and risk sharing under capitation. What is the main difference between retrospective and prospective methods of reimbursement? What are the main advantages of a prospective payment system?Regardless of the specific line of business, should all healthears businesses use the same set of ratios when conducting a financial statement analysis? Explain your answer. 17.9 What are key performance indicators (MPIs): What is a dashboggi Problems 17.1 a. Modern Medical Devices has a current ratio of 0.5. Which of the following actions would improve (i.e., increase) this ratio? . Use cash to pay off current liabilities. . Collect some of the current accounts receivable. . Use cash to pay off some long-term debt. . Purchase additional inventory on credit (i.c., accounts payable). . Sell some of the existing inventory at cost. b. Assume that the company has a current ratio of 1.2. Now which of the above actions would improve this ratio? 17.2 Southwest Physicians, a medical group practice, is just being formed. It will need $2 million of total assets to generate $3 million in revenues. Furthermore, the group expects to have a profit margin of 5 percent. The group is considering two financing alternatives. First, it can use all-equity financing by requiring each physician to contribute his or her pro rata share. Alternatively, the practice can finance up to 50 percent of its assets with a bank loan. Assuming that the debt alternative has no impact on the expected profit margin, what is the difference between the expected ROE if the group finances with 50 percent debt versus the expected ROE if it finances entirely with equity capital? 17.3 Riverside Memorial's primary financial statements are presented u exhibits 17.1, 17.2, and 17.3.5. A health intervention is performed for A. The sole purpose of preventing deaths B. The purpose of assessing, improving, maintaining, promoting or modifying health, functioning or health conditions C. With, for or on behalf of a person or a population D. The purpose of making health financing systainable E. The purpose of improving the health system 6. Primary Prevention A. Is to avoid mortality and sequelac after you have already developed the disease B. Primary prevention essentially means that environmental and behavioral risk factors, such as poor diets, air pollution, or water contamination, are being removed C. Means the individual is already exposed to risk factors, but this type of prevention aims to keep them from developing into a disease D. Could be delivered using the 3 platforms E. Addresses the behavioral and environmental factors before they become actual risk factors and lead to negative health outcomes. 7. Characteristics of the Health Service Platform A. Self regulation It. Sophisticated purchasing power C. Direct control of tner D. Intensity of transactions E. Anymincing of information s. Interventions consists of A. Human resources who provide services H. Designed for primary prevention only C. Individual programs or hymn to improve health D. Health care facilities whing services am prided E. Designed for primary, secondary and iminny prom s. The Global Burden of Disease analysis A. Routinely disaggregates the burdens of disease in each country by social variables such as wealth and education levels. B. Routinely disaggregates the burdens of disease in each country by demographic variables C. Helps identify the priority health problems in each included country, as well as their evolution D. Health interventions are currently included in the GBD E. Global burden of diseases has not changed since 1990 16. Which of the following is correct about healthcare platform: A. No supply is sequined B. Nal schedulable C. Sophisticated technology toquired D. Not self regulated E. Low ummcity of information F. Can be delivered by inal