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程序代写案例-MA318-6

时间：2020-12-16

MA318-6-AU

UNIVERSITY OF ESSEX

Final Year Examinations 2020

STATISTICAL METHODS

Time allowed: 24 hours

You are not expected to spend 24 hours on this paper, this time is to take into account personal

circumstances. You do not need to complete this paper in a single sitting. You may complete the paper at

the time(s) of day that best suit you, provided you submit by the deadline.

We expect students to be able to complete this paper within 3 hours.

IMPORTANT

If you intend to apply for IFoA exemptions you MUST submit your answers within 4 hours of the start

of the exam. FASER time stamps your submission and we will verify to the IFoA your time of

submission.

Please see your exam timetable or check on FASER for the deadline by which to upload your answers.

If you are normally allowed either additional writing time or a rest break this will not apply in this case as the

24 hours given to all students is considered sufficient. If you have any questions relating to this please

contact the Student Wellbeing and Inclusivity Team. The exception to this is if you are applying for IFoA

exemptions and you would then have your additional time on top of the standard exam length of 3 hours,

plus 1 hour.

The times shown on your timetable are in British Summer Time (BST). Please check online for a

conversion to your local time if you will be undertaking your assessment outside the United

Kingdom

Number of questions: 9

Equipment needed: Useful distribution densities at the end of this question paper

Questions are NOT of equal weight

Candidates must attempt ALL questions.

Contacts

If you believe there is an error in this paper please email maths@essex.ac.uk

If you have a technical problem with FASER, or any other query, please go to Exams Website to find

contact details of the teams that can help you.

Academic Offences

Please do not communicate with any other candidate in any way during this assessment. Your response

must be your own work. Procedures are in place to detect plagiarism and collusion.

Submitting your answers

You should scan your handwritten answers using a scanner or an app on your phone and upload to

FASER. Guidelines on the format you should use have been circulated to you previously and can be

found in the information below.

We would recommend you allow at least 30 minutes within your exam time to upload your work. Set

yourself a timer to ensure you don’t miss the deadline.

Any late submissions will receive a mark of zero.

Additional instructions on completing and submitting your exam

Format of your answers

Write your registration number clearly at the top of the first page. Do not add your name as

these will be marked anonymously.

Answers should be hand written

Write in pen – preferably black

Number your answers clearly

We recommend you start a new page for each question.

Number each page to help you check you have scanned every page

Do not write on the reverse of the page as this can show through when you scan your answers

and can affect legibility.

Scanning and uploading

If you don’t have a scanner there are also lots of apps you can use on your phone to scan your

work into a PDF format, for example Microsoft Office Lens or Scannable – but you can use

whichever one suits you. Later versions of the iPhone also have a scanning facility.

You should combine your pages into one continuous PDF document. Do NOT submit

photographs of each individual page.

Pages should be scanned in portrait orientation.

Save your document as your registration number.

Check your final document before uploading to FASER and ensure you have captured all

pages and it’s legible. If the marker cannot read your answer they will be unable to mark it.

Do not leave it until close to the deadline to upload your answers – late submissions will not be

accepted.

We recommend you check your submission after you have uploaded to ensure you have

uploaded the correct document, and retain your email receipt from FASER.

We do not anticipate any problems with FASER but if you do experience any difficulties

uploading your answers, you can email them to maths@essex.ac.uk with [MA318 exam] in the

subject. You will still need to upload to FASER but you can do this when you can access the

system. This is just a back-up, please do not email your answers if you are able to submit to

FASER.

Only your latest submission (within the deadline) will be marked.

Extenuating circumstances

If you are unable to sit this exam, you must complete a Notification of Absence form and clearly

state the reasons why you are unable to submit the work.

We know that students are working under different conditions at the moment. If you encounter any

difficulties with your exam that you believe has affected your performance you can inform the

Exam Board of this by submitting an Extenuating Circumstances form

https://www1.essex.ac.uk/students/exams-and-coursework/ext-circ.aspx

Academic Offences

You are reminded that copying from textbooks, articles or any online or offline sources is

plagiarism, which is an academic offence. Collaborating with other students is collusion, which is

also an academic offence. Any student suspected of committing an academic offence will be

interviewed in person to determine whether the work they have submitted is entirely their own.

More information about academic offences can be found at

https://www.essex.ac.uk/student/exams-and-coursework/about-academic-offences

2 MA318-6-AU

1. The table below shows the increment claims paid on a portfolio of insurance policies. Claims are

fully paid by the end of development year 3.

Policy year / Development year 0 1 2 3

2008 (year 1) 100 35 30 10

2009 (year 2) 89 20 16

2010 (year 3) 112 33

2011 (year 4) 132

We assume that the above random values are independent and from Poisson(αiβj), with αi, βj > 0,

where i is the policy year index and j is the development year index.

Use the chain ladder algorithm to estimate the values αi and βj .

[10 marks]

2. Claim amounts on a certain type of insurance policy follow an exponential distribution with

mean=5, i.e. it has probability density

f(x) = 0.2e−0.2x, x > 0.

The insurance company purchases a special type of reinsurance policy so that for a given claim X

incurred by insurer, the reinsurance company pay

0 if 0 < X < 3

0.5X − 1.5 if 3 ≤ X < 10

X − 6.5 if X ≥ 10

to the insurance company.

Calculate the expected amount paid by the reinsurance company on a random chosen claim incurred

by insurer.

[12 marks]

3. A car insurance company has a particular policy. Denote N as the number of claims per annum and

we assume that N follows a Poisson distribution with parameter µ. Claim amounts (X1, · · · , XN )

are assumed to be independent from each other, with exponential distribution λe−λx. Let

S =

∑N

i=1Xi denote the aggregate annual claims from this policy. A check is made for ruin only at

the end of the year.

Suppose that the premium income per year is c.

(a) Derive the moment generating function of N , Xi and S. [9 marks]

(b) Suppose that we can obtain the estimates for the mean and variance of S and denote them as

m1 and m2, respectively. If we use a normal approximation to the distribution of S, briefly

explain how to calculate the initial capital, U , required in order that the probability of ruin at

the end of the first year is 0.01. [3 marks]

3 MA318-6-AU

4. (a) Determine the Jeffreys non-informative prior for the parameter β of a distribution function with

density

f(x|α, β) ∝ βαxα−1e−βx

where α is supposed to be given.

[8 marks]

(b) Explain in what sense Jeffreys prior is non-informative.

[2 marks]

5. Assume that we have an independent sample X1, X2, · · · , Xn from a geometric distribution with

parameter p, where the mass function is:

f(x|p) = p(1− p)x, 0 < p ≤ 1, x = 0, 1, 2, ...

The prior density function for p is given by

f(p) =

Γ(8)

Γ(5)Γ(3)

p4(1− p)2, 0 < p < 1

a. Find the posterior distribution for p. [4 marks]

b. Show that, under the prior distribution, E

(

1−p

p

)

= 3

4

. [8 marks]

6. Suppose the weather of a Sunday can be one of three possibilities, θ1 (rainy), θ2 (windy) or θ3

(sunny) and a person can choose one of the three actions to do on that day a1 (picnic), a2 (shopping)

or a3 (waching football match). Through consideration, he arrives at the following loss matrix

a1 a2 a3

θ1 4 1 2

θ2 0 1 5

θ3 1 2 0

Suppose an action δ of the person must have the form δ = p1 < a1 > +p2 < a2 > +p3 < a3 >,

which means with probability pk the person takes action ak, k = 1, 2, 3.

Find the best δ based on the minimax rule.

[15 marks]

7. Suppose that we observe one data point x ∼ N(θ, σ2) where σ2 is known. The prior distribution is

θ ∼ N(0, 1). Find the (prior) predictive distribution.

[12 marks]

4 MA318-6-AU

8. The waiting time for a bus at a given station at a certain time of a day is known to have a uniform,

U(0, θ), distribution. It is desired to test H0 : 0 < θ < 12 versus H1 : θ > 12. From other similar

routes, it is known that θ has a Pareto distribution P(5, 3). If the waiting times 10, 3, 2, 5, 8 are

observed, calculate the posterior probability of each hypothesis and the Bayes factor.

(You are given the following information:

The Pareto P(x0, α) distribution has a pdf,

f(θ|x0, α) = α

x0

(x0

θ

)α+1

I(x0,∞)(θ)

with θ > x0, 0 < x0 <∞ and α > 0. It has mean αx0/(α− 1) if α > 1.)

[12 marks]

9. Suppose we obtain a posterior which is a Beta(5,8) distribution. The following R code estimates the

mean of this posterior distribution. Explain each step of the following R code and explain why it

provides a consistent estimate for the mean of this posterior.

n = 1000

alpha = 5

beta =8

x = rbeta(n, alpha, beta)

mean(x)

[5 marks]

END OF PAPER

MA318 – useful distribution densities

1 p follows a Beta-distribution, Beta(; ), then it has probability density function (pdf)

f(pj; ) = 1

B(; )

p1(1 p)1; p 2 [0; 1]; ; > 0

with mean =( + ). Here

B(; ) =

()()

(+ )

and () has the following property

( + 1) = ():

2 p = (p1; ; pk) follows a Dirichlet-distribution, Dirichlet(1; ; k), then it has pdf

f(p1; ; pkj1; ; k) = (

P

i i)

(1) (k)p

11

1 pk1k ;

X

pi = 1; pi 2 [0; 1]; i > 0

with mean k= (

P

i i) for pk.

3 X follows a Binomial-distribution, then it has pdf

f(xjn; p) =

n

x

px(1 p)nx; p 2 [0; 1]

with mean np.

4 follows a Normal distribution of mean and variance 2, then it has pdf

f(j; 2) = 1p

22

e

()2

22

2 MA318 – useful distribution densities

5 X follows a Negative binomial (NB) distribution, NB(r; p), then it has pdf

f(xjr; p) =

x+ r 1

x

(1 p)rpx

Note that the NB random variable X means, in a sequence of i.i.d. Bernoulli trials, the number of

successes before r failures occur.

6 X = (X1; ; Xk) follows a multinomial distribution,M(n;p), then it has pdf

f(xjn;p) = n!

x1! xk!p

x1

1 pxkk

where

P

i xi = n, pi 2 [0; 1] and

P

i pi = 1.

7 follows a Gamma distribution, Gamma(; ), then it has pdf

f(j; ) =

()

1e; ; > 0

Note that if = 1, the gamma distribution becomes an exponential distribution.

8 X follows a uniform distribution, Uniform(a; b) (or using notation U [a; b], uniform[a; b]), then it has

pdf

f(xja; b) = 1

b a; for all x 2 [a; b];

otherwise f(xja; b) = 0.

9 X follows a Poisson(), then it has probability mass function

p(xj) =

x

x!

e;

UNIVERSITY OF ESSEX

Final Year Examinations 2020

STATISTICAL METHODS

Time allowed: 24 hours

You are not expected to spend 24 hours on this paper, this time is to take into account personal

circumstances. You do not need to complete this paper in a single sitting. You may complete the paper at

the time(s) of day that best suit you, provided you submit by the deadline.

We expect students to be able to complete this paper within 3 hours.

IMPORTANT

If you intend to apply for IFoA exemptions you MUST submit your answers within 4 hours of the start

of the exam. FASER time stamps your submission and we will verify to the IFoA your time of

submission.

Please see your exam timetable or check on FASER for the deadline by which to upload your answers.

If you are normally allowed either additional writing time or a rest break this will not apply in this case as the

24 hours given to all students is considered sufficient. If you have any questions relating to this please

contact the Student Wellbeing and Inclusivity Team. The exception to this is if you are applying for IFoA

exemptions and you would then have your additional time on top of the standard exam length of 3 hours,

plus 1 hour.

The times shown on your timetable are in British Summer Time (BST). Please check online for a

conversion to your local time if you will be undertaking your assessment outside the United

Kingdom

Number of questions: 9

Equipment needed: Useful distribution densities at the end of this question paper

Questions are NOT of equal weight

Candidates must attempt ALL questions.

Contacts

If you believe there is an error in this paper please email maths@essex.ac.uk

If you have a technical problem with FASER, or any other query, please go to Exams Website to find

contact details of the teams that can help you.

Academic Offences

Please do not communicate with any other candidate in any way during this assessment. Your response

must be your own work. Procedures are in place to detect plagiarism and collusion.

Submitting your answers

You should scan your handwritten answers using a scanner or an app on your phone and upload to

FASER. Guidelines on the format you should use have been circulated to you previously and can be

found in the information below.

We would recommend you allow at least 30 minutes within your exam time to upload your work. Set

yourself a timer to ensure you don’t miss the deadline.

Any late submissions will receive a mark of zero.

Additional instructions on completing and submitting your exam

Format of your answers

Write your registration number clearly at the top of the first page. Do not add your name as

these will be marked anonymously.

Answers should be hand written

Write in pen – preferably black

Number your answers clearly

We recommend you start a new page for each question.

Number each page to help you check you have scanned every page

Do not write on the reverse of the page as this can show through when you scan your answers

and can affect legibility.

Scanning and uploading

If you don’t have a scanner there are also lots of apps you can use on your phone to scan your

work into a PDF format, for example Microsoft Office Lens or Scannable – but you can use

whichever one suits you. Later versions of the iPhone also have a scanning facility.

You should combine your pages into one continuous PDF document. Do NOT submit

photographs of each individual page.

Pages should be scanned in portrait orientation.

Save your document as your registration number.

Check your final document before uploading to FASER and ensure you have captured all

pages and it’s legible. If the marker cannot read your answer they will be unable to mark it.

Do not leave it until close to the deadline to upload your answers – late submissions will not be

accepted.

We recommend you check your submission after you have uploaded to ensure you have

uploaded the correct document, and retain your email receipt from FASER.

We do not anticipate any problems with FASER but if you do experience any difficulties

uploading your answers, you can email them to maths@essex.ac.uk with [MA318 exam] in the

subject. You will still need to upload to FASER but you can do this when you can access the

system. This is just a back-up, please do not email your answers if you are able to submit to

FASER.

Only your latest submission (within the deadline) will be marked.

Extenuating circumstances

If you are unable to sit this exam, you must complete a Notification of Absence form and clearly

state the reasons why you are unable to submit the work.

We know that students are working under different conditions at the moment. If you encounter any

difficulties with your exam that you believe has affected your performance you can inform the

Exam Board of this by submitting an Extenuating Circumstances form

https://www1.essex.ac.uk/students/exams-and-coursework/ext-circ.aspx

Academic Offences

You are reminded that copying from textbooks, articles or any online or offline sources is

plagiarism, which is an academic offence. Collaborating with other students is collusion, which is

also an academic offence. Any student suspected of committing an academic offence will be

interviewed in person to determine whether the work they have submitted is entirely their own.

More information about academic offences can be found at

https://www.essex.ac.uk/student/exams-and-coursework/about-academic-offences

2 MA318-6-AU

1. The table below shows the increment claims paid on a portfolio of insurance policies. Claims are

fully paid by the end of development year 3.

Policy year / Development year 0 1 2 3

2008 (year 1) 100 35 30 10

2009 (year 2) 89 20 16

2010 (year 3) 112 33

2011 (year 4) 132

We assume that the above random values are independent and from Poisson(αiβj), with αi, βj > 0,

where i is the policy year index and j is the development year index.

Use the chain ladder algorithm to estimate the values αi and βj .

[10 marks]

2. Claim amounts on a certain type of insurance policy follow an exponential distribution with

mean=5, i.e. it has probability density

f(x) = 0.2e−0.2x, x > 0.

The insurance company purchases a special type of reinsurance policy so that for a given claim X

incurred by insurer, the reinsurance company pay

0 if 0 < X < 3

0.5X − 1.5 if 3 ≤ X < 10

X − 6.5 if X ≥ 10

to the insurance company.

Calculate the expected amount paid by the reinsurance company on a random chosen claim incurred

by insurer.

[12 marks]

3. A car insurance company has a particular policy. Denote N as the number of claims per annum and

we assume that N follows a Poisson distribution with parameter µ. Claim amounts (X1, · · · , XN )

are assumed to be independent from each other, with exponential distribution λe−λx. Let

S =

∑N

i=1Xi denote the aggregate annual claims from this policy. A check is made for ruin only at

the end of the year.

Suppose that the premium income per year is c.

(a) Derive the moment generating function of N , Xi and S. [9 marks]

(b) Suppose that we can obtain the estimates for the mean and variance of S and denote them as

m1 and m2, respectively. If we use a normal approximation to the distribution of S, briefly

explain how to calculate the initial capital, U , required in order that the probability of ruin at

the end of the first year is 0.01. [3 marks]

3 MA318-6-AU

4. (a) Determine the Jeffreys non-informative prior for the parameter β of a distribution function with

density

f(x|α, β) ∝ βαxα−1e−βx

where α is supposed to be given.

[8 marks]

(b) Explain in what sense Jeffreys prior is non-informative.

[2 marks]

5. Assume that we have an independent sample X1, X2, · · · , Xn from a geometric distribution with

parameter p, where the mass function is:

f(x|p) = p(1− p)x, 0 < p ≤ 1, x = 0, 1, 2, ...

The prior density function for p is given by

f(p) =

Γ(8)

Γ(5)Γ(3)

p4(1− p)2, 0 < p < 1

a. Find the posterior distribution for p. [4 marks]

b. Show that, under the prior distribution, E

(

1−p

p

)

= 3

4

. [8 marks]

6. Suppose the weather of a Sunday can be one of three possibilities, θ1 (rainy), θ2 (windy) or θ3

(sunny) and a person can choose one of the three actions to do on that day a1 (picnic), a2 (shopping)

or a3 (waching football match). Through consideration, he arrives at the following loss matrix

a1 a2 a3

θ1 4 1 2

θ2 0 1 5

θ3 1 2 0

Suppose an action δ of the person must have the form δ = p1 < a1 > +p2 < a2 > +p3 < a3 >,

which means with probability pk the person takes action ak, k = 1, 2, 3.

Find the best δ based on the minimax rule.

[15 marks]

7. Suppose that we observe one data point x ∼ N(θ, σ2) where σ2 is known. The prior distribution is

θ ∼ N(0, 1). Find the (prior) predictive distribution.

[12 marks]

4 MA318-6-AU

8. The waiting time for a bus at a given station at a certain time of a day is known to have a uniform,

U(0, θ), distribution. It is desired to test H0 : 0 < θ < 12 versus H1 : θ > 12. From other similar

routes, it is known that θ has a Pareto distribution P(5, 3). If the waiting times 10, 3, 2, 5, 8 are

observed, calculate the posterior probability of each hypothesis and the Bayes factor.

(You are given the following information:

The Pareto P(x0, α) distribution has a pdf,

f(θ|x0, α) = α

x0

(x0

θ

)α+1

I(x0,∞)(θ)

with θ > x0, 0 < x0 <∞ and α > 0. It has mean αx0/(α− 1) if α > 1.)

[12 marks]

9. Suppose we obtain a posterior which is a Beta(5,8) distribution. The following R code estimates the

mean of this posterior distribution. Explain each step of the following R code and explain why it

provides a consistent estimate for the mean of this posterior.

n = 1000

alpha = 5

beta =8

x = rbeta(n, alpha, beta)

mean(x)

[5 marks]

END OF PAPER

MA318 – useful distribution densities

1 p follows a Beta-distribution, Beta(; ), then it has probability density function (pdf)

f(pj; ) = 1

B(; )

p1(1 p)1; p 2 [0; 1]; ; > 0

with mean =( + ). Here

B(; ) =

()()

(+ )

and () has the following property

( + 1) = ():

2 p = (p1; ; pk) follows a Dirichlet-distribution, Dirichlet(1; ; k), then it has pdf

f(p1; ; pkj1; ; k) = (

P

i i)

(1) (k)p

11

1 pk1k ;

X

pi = 1; pi 2 [0; 1]; i > 0

with mean k= (

P

i i) for pk.

3 X follows a Binomial-distribution, then it has pdf

f(xjn; p) =

n

x

px(1 p)nx; p 2 [0; 1]

with mean np.

4 follows a Normal distribution of mean and variance 2, then it has pdf

f(j; 2) = 1p

22

e

()2

22

2 MA318 – useful distribution densities

5 X follows a Negative binomial (NB) distribution, NB(r; p), then it has pdf

f(xjr; p) =

x+ r 1

x

(1 p)rpx

Note that the NB random variable X means, in a sequence of i.i.d. Bernoulli trials, the number of

successes before r failures occur.

6 X = (X1; ; Xk) follows a multinomial distribution,M(n;p), then it has pdf

f(xjn;p) = n!

x1! xk!p

x1

1 pxkk

where

P

i xi = n, pi 2 [0; 1] and

P

i pi = 1.

7 follows a Gamma distribution, Gamma(; ), then it has pdf

f(j; ) =

()

1e; ; > 0

Note that if = 1, the gamma distribution becomes an exponential distribution.

8 X follows a uniform distribution, Uniform(a; b) (or using notation U [a; b], uniform[a; b]), then it has

f(xja; b) = 1

b a; for all x 2 [a; b];

otherwise f(xja; b) = 0.

9 X follows a Poisson(), then it has probability mass function

p(xj) =

x

x!

e;