DSC4821_2022_Assignment_03

pdf

School

University of South Africa *

*We aren’t endorsed by this school

Course

4821

Subject

Statistics

Date

Nov 24, 2024

Type

pdf

Pages

Uploaded by ChancellorFang8501

DSC4821: Questions for Assignment 03 Question 1 Consider the Markov chain whose transition probability matrix is: P =         0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 3 1 3 1 3 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 1 2         (a) Classify the states { 0 , 1 , 2 , 3 , 4 , 5 } into classes. (b) Identify the recurrent and transient classes of (a). Question 2 In a Markov chain model for the progression of a disease, X n denotes the level of severity in year n , for n = 0 , 1 , 2 , 3 , . . . . The state space is { 1 , 2 , 3 , 4 } with the following interpretations: in state 1 the symptoms are under control, state 2 represents moderate symptoms, state 3 represents severe symptoms and state 4 represents a permanent disability. The transition matrix is: P =     1 4 1 2 0 1 4 0 1 4 1 2 1 4 0 0 1 2 1 2 0 0 0 1     (a) Classify the four states as transient or recurrent giving reasons. What does this tell you about the long-run fate of someone with this disease? (b) Calculate the 2-step transition matrix. (c) Determine ( i ) the probability that a patient whose symptoms are moderate will be permanently disabled two years later and ( ii ) the probability that a patient whose symptoms are under control will have severe symptoms one year later. (d) Calculate the probability that a patient whose symptoms are moderate will have severe symptoms four years later. A new treatment becomes available but only to permanently disabled patients, all of whom receive the treatment. This has a 75% success rate in which case a patient returns to the “symptoms under control” state and is subject to the same transition probabilities as before. A patient whose treatment is unsuccessful remains in state 4 receiving a further round of treatment the following year. (e) Write out the transition matrix for this new Markov chain and classify the states as transient or recurrent. (f) Calculate the stationary distribution of the new chain. (g) The annual cost of health care for each patient is 0 in state 1, $1 000 in state 2, $2 000 in state 3 and $8 000 in state 4. Calculate the expected annual cost per patient when the system is in steady state. 1

Question 3 Let { X n , n ≥ 1 } be a sequence of independent and identically distributed random vari- ables such that P { X n = 1 } = 1 3 and P { X n = - 1 } = 2 3 . Define S n = X 1 + X 2 + . . . + X n and set F n = σ ( X 1 , X 2 , . . . , X n ). Show that the process ( M n ) defined by M n = S n + n 3 is a martingale but the process ( S n ) is not a martingale. Question 4 Suppose that a fisherman catches fish according to a Poisson process with rate 2 per hour. We know that yesterday he began fishing at 9:00 am. (a) What is the probability that he caught exactly two fish by 11:00 am yesterday? (b) What is the probability that he caught exactly two fish by 11:00 am yesterday and five fish by 11:30 am yesterday? (c) What is the expected time that he caught his fifth fish? (d) Suppose that we know that by 1:00 pm yesterday he caught exactly three fish. What is the probability that by 2:00 pm yesterday he caught a total of exactly ten fish? And what is the probability that he caught his first fish after 10:00 am yesterday? (e) Suppose that we know that he didn’t catch any fish until after 10:00 am yesterday. Given that information, what is the expected time that he caught his first fish? Question 5 At an underground station, trains arrive according to a Poisson process of rate twenty per hour. (a) Suppose that I arrive at the platform and intend to take the first westbound train that arrives. What is the expected time I have to wait until the first train arrives? (b) After ten minutes, no train has arrived and I am still on the platform. What is the expected further time I have to wait until the first train arrives? Question 6 In this question assume that { B ( t ) : t ≥ 00 } is the standard Brownian motion (BM). (a) Find P { B (1) + B (2) > 2 } . (b) Write a formula for the conditional p.d.f. of B (2) given B (1) = 3. (c) Write a formula for the conditional p.d.f. of B (1) given B (2) = 3. d) Using conditions defining the BM, give a detailed proof of the Brownian scaling property: { X ( t ) : t ≥ 0 } defined by X ( t ) = √ 2 B ( t/ 2) is also a BM. (f) For the geometric BM { X ( t ) = e B ( t ) , t ≥ 0 } , calculate E [ X (2)] and V ar [ X (2)]. Question 7 In this question assume that { B ( t ) , t ≥ 0 } is the standard Brownian motion (BM). (a) The Brownian bridge on [0,1] is defined as X ( t ) = B ( t ) - tB (1) , t ∈ [0 , 1] . This process is clearly a Gaussian process. Find its mean and covariance functions. (b) With the Brownian bridge defined in question (a), show that the process { W ( t ) : t ∈ [0 , 1] } defined by W ( t ) = (1 + t ) X t 1 + t , 0 ≤ t ≤ 1 is a Brownian motion on the interval [0 , 1]. Question 8 Let { B ( t ) : t ≥ 0 } be a standard Brownian motion with B (0) = 0. 2

(a) Given t > 0, state the distribution of B ( t ), including any parameters. (b) Determine the joint probability density function f B (1) ,B (2) ( x 1 , x 2 ) . Question 9 Let ( X 1 , X 2 , . . . , X n , . . . ) be a sequence of independent random variables with zero mean and variance V ar [ X n ] = σ 2 n . Let S n = X 1 + X 2 + . . . + X n and T 2 n = σ 2 1 + σ 2 2 + . . . + σ 2 n . Show that S 2 n - T 2 n : n = 1 , 2 , . . . is a martingale. Question 10 A radioactive source emits particles according to a Poisson process with rate two particles per minute. (a) What is the probability that the first particle appears after three minutes? (b) What is the probability that the first particle appears after three minutes but before five minutes? (c) What is the probability that exactly one particle is emitted in the interval from three to five minutes? (d) What is the probability that exactly one particle is emitted in the interval from zero to four minutes and that exactly one particle is emitted in the interval from three to five minutes? Question 11 Let H be a real number such that 0 < H < 1. A Gaussian process { X ( t ) : t ≥ 0 } is called a fractional Brownian motion with Hurst parameter H if ( i ) X (0) = 0, ( ii ) X ( t ) is normally distributed with mean 0 and variance t 2 H , ( iii ) { X ( t ) : t ≥ 0 } has stationary increments. (a) Prove that for a fractional Brownian motion { X ( t ) : t ≥ 0 } , the autocovariance function is given by K X ( t, s ) = E [ X ( t ) X ( s )] = 1 2 ( t 2 H + s 2 H - | t - s | 2 H ) . (b) Prove that for H = 1 / 2, a fractional Brownian motion has also independent incre- ments and hence it is a Brownian motion but for H 6 = 1 / 2, a fractional Brownian motion does not have independent increments. Question 12 A certain supply company describes its receivable accounts (debts) as follows: – State P: A debt is in state P if it has been paid. – State C: A dept is in state C if it is current (less than a month old). – State I: A debt is in state I if it is one month old – State B: A debt is in state B if it is at least 2 months old. Debts in state B are listed as bad debts. An analyst considers historical data and model the dynamics of the company accounts as a Markov chain with the following one-month transition matrix (with respect to the states P, B, C, I in that order): T =     1 . 00 0 . 00 0 . 00 0 . 00 0 . 00 1 . 00 0 . 00 0 . 00 0 . 70 0 . 00 0 . 00 0 . 30 0 . 35 0 . 65 0 . 00 0 . 00     3

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Once a debt is paid (i.e., once the item enters state P), the probability of moving to state B, C, or I is obviously zero. If a debt is in state C at the beginning of the month, then at the end of the month, there is a 0.70 probability that it will state P and a 0.30 probability that it will enter state I. If a debt is in state I at the beginning of the month, then at the end of the month, there is a 0.35 probability that it will enter state P and a 0.65 probability that it will become a bad debt. Finally after an account has been listed as a bad account (that is, state B), it is transferred to the company overdue accounts section for collection. (a) Classify the four states as transient, recurrent or absorbing states, giving reasons. (b) Calculate the 2-step transition matrix. (c) Calculate the fundamental matrix Φ of the Markov chain. (d) Under normal circumstances, it is assumed the store averages R2 300 000 in outstand- ing debts during an average month; R1 500 000 of this amount is current and R800 000 is 1 month old. Determine how much of this amount will eventually be paid or end up as bad debts in a typical month. Hint: To answer this question, consider ( 1 500 000 800 000 ) Φ R where R = 0 . 70 0 . 00 0 . 35 0 . 65 where Φ is the fundamental matrix of the Markov chain (see relation (5.50), page 215 in the prescribed book). The answer is of the form ( a, b ) where a is the amount that will be paid while the amount b will be declared as bad dept Question 13 (Background) As a result of intense competition and an economic recession, Davidsons Department Store in Atlanta was forced to pay particularly close attention to its cash flow. Because of the poor economy, a number of Davidsons customers were not paying their bills upon receipt,delaying payment for several months,and frequently not paying at all. In general, the Davidsons policy for accounts receivable was to allow a customer to be 2 months late on his or her bill before turning it over to a collection agency. However,it was not quite as simple as that. Davidsons has approximately 10,000 open accounts at any time. The age of the account is determined by the oldest dollar owed. This means that a customer can have a balance for items bought in two different months, with the overall account being listed as old as the earliest month of purchase. For example, suppose a customer has a balance of $100 at the end of January, $80 of which is for items bought in January and $20 for items bought in November. This means the account is 2 months old at the end of January because the oldest amount on account is from November. If the customer subsequently pays $20 on the bill in February,this cancels the November purchase.Then if the customer makes $100 worth of purchases in February, the account is $180, and it is 1 month old (since the oldest purchases were from January). (Question) Carla Reata, Davidson’s comptroller, analyzed the accounts receivable data for the store for an extended period. She summarized these data and developed some probabilities for the payment (or nonpayment) of bills. She determined that for current bills (in their first month of billing), there is an 0.86 probability that the bills will be paid in the month and a 0.14 probability that they will be carried over to the next month and be 1 month late. If a bill is already 1 month late, there is a 0.22 probability that the oldest portion of the bill will be paid so that it will remain 1 month old, a 0.46 probability that the entire bill will be carried over so that it is 2 months old,and a 0.32 probability that 4

Table 1: Table 1: One-year transition probabilities matrix Ratings at year-end Initial ratings AAA AA A BBB BB B CCC Default AAA 0.9366 0.0583 0.0040 0.0009 0.0002 0 0 0 AA 0.0066 0.9172 0.0694 0.0049 0.0006 0.0009 0.0002 0.0002 A 0.0007 0.0225 0.9176 0.0518 0.0049 0.0020 0.0001 0.0004 BBB 0.0003 0.0026 0.0483 0.8924 0.0444 0.0081 0.0016 0.0023 BB 0.0003 0.0006 0.0044 0.0666 0.8323 0.0746 0.0105 0.0107 B 0 0.0010 0.0032 0.0046 0.0572 0.8362 0.0384 0.0594 CCC 0.0015 0 0.0029 0.0088 0.0191 0.1028 0.6123 0.2526 Default 0 0 0 0 0 0 0 1.0000 the bill will be paid in the month.For bills 2 months old,there is a probability of 0.54 that the oldest portion will be paid so that the bill remains 1 month old, a 0.16 probability that the next-oldest portion of the bill will be paid so that it remains 2 months old, a 0.18 probability that the bill will be paid in the month, and a 0.12 probability that the bill will be listed as a bad debt and turned over to a collection agency. If a bill is paid or listed as a bad debt, it will no longer move to any other billing status. Under normal circumstances (i.e.,not a holiday season),the store averages $1,350,000 in outstanding bills during an average month; $750,000 of this amount is current, $400,000 is 1 month old, and $200,000 is 2 months old. The vice president of finance for the store wants Carla to determine how much of this amount will eventually be paid or end up as bad debts in a typical month. She also wants Carla to tell her if an average cash reserve of $60,000 per month is enough to cover the expected bad debts that will occur each month. Perform this analysis for Carla. Hint: First determine the transition matrix and follow the same argument as in the pre- vious question to answer this question. Question 14 In modeling insured automobile drivers’ ratings by the insurer, you might want to consider states such as Preferred, Standard, and Substandard. Models describe the probabilities of moving back and forth among these states. Consider a driver-ratings model in which drivers move among the classifications Preferred, Standard, and Substandard at the end of each year. Each year: 60% of Preferreds are reclassified as Preferred, 30% as Standard, and 10% as substandard; 50% of Standards are reclassified as Standard, 30% as Preferred, and 20% as Substandard; and 60% of Substandards are reclassified as Substandard, 40% as Standard, and 0% as Preferred. (a) Show that the probability that a driver, classified as Standard at the start of the first year, will be classified as Standard at the start of the fourth year is 0.409. (b) Show that the probability that a driver, classified as Standard at the start of the first year, will be classified as Standard at the start of each of the first four years is 0.125. Question 15 Over time, bonds are liable to move from one rating category to another. This is sometimes referred to as credit ratings migration . Rating agencies produce from historical data a rating transition matrix. This matrix shows the probabiltiy of a bond moving from one rating to another during a certain period of time. Usually the period of time is one year. Below is a table giving a rating transition matrix produced from historical data by Standard and Poor’s (S&P) (Source: Standard & Poor’s, January 2001) 5

In reality, this transition matrix is updated every year. However if assuming no significant change in the transition matrix in the future, then one can use the transition matrix to predict what will happen over several years in the future. In particular, one can regard the transition matrix as a specification of a Markov chain model. Answer the following questions (a) Classify the states of ratings (transient or recurrent). What ratings are absorbing states? (b) Determine the classes of communicating states. (c) What is the probability that a currently “AAA” rated bond becomes default after 4 years? (d) Now consider what will happen in the long run, assuming that the transition matrix above operates every year. In the long run, what fraction of bonds are in “AAA”? and what fraction of bonds are in default? 6

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

DSC4821_2022_Assignment_03

Related Documents