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程序代写案例-MA318-7

时间：2020-12-16

MA318-7-AU

UNIVERSITY OF ESSEX

Graduate Examinations 2019

STATISTICAL METHODS

Time allowed: 3 hours

The paper consists of TEN questions.

Candidates must answer ALL questions.

The questions are NOT of equal weight.

Useful distribution densities are provided at the end of the question paper.

Candidates are permitted to bring into the examination room:

Scientific calculator – one of the following models only:

Casio FX-83GT PLUS

Casio FX-85GT PLUS

Please do not leave your seat unless you are given permission by an Invigilator. Do not

communicate in any way with any other candidate in the examination room. Do not open the

question paper until told to do so. All answers must be written in the answer book(s)

provided. All rough work must be written in the answer book(s) provided. A line should be

drawn through any rough work to indicate to the examiner that it is not part of the work to be

marked.

At the end of the examination, remain seated until your answer book(s) have been collected

and you have been told you may leave.

2 MA318-7-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

2015 (year 1) 103 32 29 13

2016 (year 2) 88 21 16

2017 (year 3) 110 35

2018 (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.

Calculate the estimates for the values αi and βj via the chain ladder algorithm.

[10 marks]

2. An office worker receives a random number of emails each day. The number of emails per day

follows a Poisson distribution with unknown parameter µ. Prior beliefs about µ are specified by a

gamma distribution with parameter a and b, i.e. the prior density function is

pi(µ) =

ba

Γ(a)

µa−1e−bµ.

Suppose that we have an observation x. Calculate the posterior distribution for µ and name the

posterior distribution.

[6 marks]

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

50, i.e. with pdf

0.02e−0.02x, x > 0.

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

the reinsurance company pays

0 if 0 < X < 30

0.5X − 15 if 30 ≤ X < 100

X − 65 if X ≥ 100

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

[12 marks]

3 MA318-7-AU

4. Consider the binomial model P [X = x|p] = (n

x

)

px(1− p)n−x. Suppose that n is known. Answer the

following questions with a uniform prior distribution for p.

(a) Write down the posterior distribution for p, with a single observation x. [3 marks]

(b) Calculate the prior predictive distribution. [4 marks]

(c) Now we consider the binomial model as n Bernoulli trials with success probability p. If we

observed an observation X = x1, which means x1 successes in n trials with each trail having

success probability p, calculate the posterior predictive distribution for the next trial to be

success. You may use the following result in your calculation,∫ 1

0

pα−1(1− p)β−1dp = (α− 1)!(β − 1)!

(α + β − 1)! .

[5 marks]

5. Suppose that X and Y are two independent random variables and each follows the standard normal

distribution. Use the convolution formula to find the distribution of Z = X + Y . [8 marks]

6. A Physician diagnoses a disease as being one of three possibilites, θ1, θ2 or θ3, and can prescribe

medicine a1, a2 or a3. Through consideration of the seriousness of each disease and the effectiveness

of the various medicines, he arrives at the following loss matrix

a1 a2 a3

θ1 7 1 3

θ2 0 1 6

θ3 1 2 0

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

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

Find the best decision δ based on the minimax rule. [15 marks]

4 MA318-7-AU

7. A doctor in consultation must recommend a drug dosage level for a patient. There is an unknown

minimal dosage, 0 ≤ θ ≤ 1, which will affect a cure, and the drug has serious deleterious side

effects if the dosage level is too high. Letting a denote the dosage level, the loss functions of the

doctor is of the form

L(θ, a) = 2I(a,1)(θ) + (a− θ)I(0,a)(θ).

The opinion of the doctor concerning θ (after weighting all evidence and discussing the situation), is

reflected by the density pi(θ) = 2(1− θ) (on (0, 1), of course).

Find the Bayes estimator for a. [10 marks]

8. (a) An insurance company has a set of n risks (i = 1, · · · , n) for which it has recorded the number

of claims per year, Yij , for m years (j = 1, · · · ,m).

Suppose the Yij (j = 1, · · · ,m) follows a Poisson distribution with mean exp(βi). Explain

how to find the maximum likelihood estimate for βi.

[3 marks]

(b) State the Central Limit Theorem. [3 marks]

9. An insurance company sells type I car insurance policies for major accidents. Denote N(t) as the

number of type I claims by time t, which follows a Poisson process with rate η. Denote

Xi, i = 1, · · · , N(t) as each claim amounts, which are independent and are exponentially distributed

with mean λ. The random variables Xis are also independent of N(t).

This insurance company also sells type II car insurance policies for minor accidents. Denote Y (t) as

the number of type II claims by time t, which follows a Poisson process with rate δ. For each type II

claim, the claim amount is fixed, having value A.

Let S(t) denote the aggregate claims from the portfolio of the two types of policies, i.e.

S(t) =

∑N(t)

i=1 Xi + AY (t). Show that the mean and variance of S(t) are given by

E(S(t)) = ληt+ Aδt

and

V ar(S(t)) = ηt · (2λ2) + A2δt

[14 marks]

5 MA318-7-AU

10. It is known that the random variable X has probability density function of the form

f(x|θ) =

{

1

θ

e−x/θ, 0 < x <∞

0 otherwise

The sample is X1, X2 of size n = 2.

Suppose that we want to test the hypothesis H0 : θ = 2 against H1 : θ = 4. We choose the critical

region as C = {(x1, x2) : 5 ≤ (x1 + x2)/2 <∞}. Determine the significance level.

[7 marks]

END OF PAPER

6 MA318 – useful distribution densities

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

f(p|α, β) = 1

B(α, β)

pα−1(1− p)β−1, p ∈ [0, 1], α, β > 0

with mean α/(α + β). Here

B(α, β) =

Γ(α)Γ(β)

Γ(α + β)

and Γ(α) has the following property

Γ(α + 1) = αΓ(α).

2 p = (p1, · · · , pk) follows a Dirichlet-distribution, Dirichlet(α1, · · · , αk), then it has pdf

f(p1, · · · , pk|α1, · · · , αk) = Γ(

∑

i αi)

Γ(α1) · · ·Γ(αk)p

α1−1

1 · · · pαk−1k ,

∑

pi = 1, pi ∈ [0, 1], αi > 0

with mean αk/ (

∑

i αi) for pk.

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

f(x|n, p) =

(

n

x

)

px(1− p)n−x, p ∈ [0, 1]

with mean np.

4 θ follows a Normal distribution of mean µ and variance σ2, then it has pdf

f(θ|µ, σ2) = 1√

2piσ2

e−

(θ−µ)2

2σ2

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

f(x|r, 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(x|n,p) = n!

x1! · · ·xk!p

x1

1 · · · pxkk

where

∑

i xi = n, pi ∈ [0, 1] and

∑

i pi = 1.

7 MA318 – useful distribution densities

7 λ follows a Gamma distribution, Gamma(α, β), then it has pdf

f(λ|α, β) = β

α

Γ(α)

λα−1e−βλ, α, β > 0

Note that if α = 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(x|a, b) = 1

b− a, for all x ∈ [a, b];

otherwise f(x|a, b) = 0.

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

p(x|λ) = λ

x

x!

e−λ,

UNIVERSITY OF ESSEX

Graduate Examinations 2019

STATISTICAL METHODS

Time allowed: 3 hours

The paper consists of TEN questions.

Candidates must answer ALL questions.

The questions are NOT of equal weight.

Useful distribution densities are provided at the end of the question paper.

Candidates are permitted to bring into the examination room:

Scientific calculator – one of the following models only:

Casio FX-83GT PLUS

Casio FX-85GT PLUS

Please do not leave your seat unless you are given permission by an Invigilator. Do not

communicate in any way with any other candidate in the examination room. Do not open the

question paper until told to do so. All answers must be written in the answer book(s)

provided. All rough work must be written in the answer book(s) provided. A line should be

drawn through any rough work to indicate to the examiner that it is not part of the work to be

marked.

At the end of the examination, remain seated until your answer book(s) have been collected

and you have been told you may leave.

2 MA318-7-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

2015 (year 1) 103 32 29 13

2016 (year 2) 88 21 16

2017 (year 3) 110 35

2018 (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.

Calculate the estimates for the values αi and βj via the chain ladder algorithm.

[10 marks]

2. An office worker receives a random number of emails each day. The number of emails per day

follows a Poisson distribution with unknown parameter µ. Prior beliefs about µ are specified by a

gamma distribution with parameter a and b, i.e. the prior density function is

pi(µ) =

ba

Γ(a)

µa−1e−bµ.

Suppose that we have an observation x. Calculate the posterior distribution for µ and name the

posterior distribution.

[6 marks]

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

50, i.e. with pdf

0.02e−0.02x, x > 0.

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

the reinsurance company pays

0 if 0 < X < 30

0.5X − 15 if 30 ≤ X < 100

X − 65 if X ≥ 100

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

[12 marks]

3 MA318-7-AU

4. Consider the binomial model P [X = x|p] = (n

x

)

px(1− p)n−x. Suppose that n is known. Answer the

following questions with a uniform prior distribution for p.

(a) Write down the posterior distribution for p, with a single observation x. [3 marks]

(b) Calculate the prior predictive distribution. [4 marks]

(c) Now we consider the binomial model as n Bernoulli trials with success probability p. If we

observed an observation X = x1, which means x1 successes in n trials with each trail having

success probability p, calculate the posterior predictive distribution for the next trial to be

success. You may use the following result in your calculation,∫ 1

0

pα−1(1− p)β−1dp = (α− 1)!(β − 1)!

(α + β − 1)! .

[5 marks]

5. Suppose that X and Y are two independent random variables and each follows the standard normal

distribution. Use the convolution formula to find the distribution of Z = X + Y . [8 marks]

6. A Physician diagnoses a disease as being one of three possibilites, θ1, θ2 or θ3, and can prescribe

medicine a1, a2 or a3. Through consideration of the seriousness of each disease and the effectiveness

of the various medicines, he arrives at the following loss matrix

a1 a2 a3

θ1 7 1 3

θ2 0 1 6

θ3 1 2 0

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

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

Find the best decision δ based on the minimax rule. [15 marks]

4 MA318-7-AU

7. A doctor in consultation must recommend a drug dosage level for a patient. There is an unknown

minimal dosage, 0 ≤ θ ≤ 1, which will affect a cure, and the drug has serious deleterious side

effects if the dosage level is too high. Letting a denote the dosage level, the loss functions of the

doctor is of the form

L(θ, a) = 2I(a,1)(θ) + (a− θ)I(0,a)(θ).

The opinion of the doctor concerning θ (after weighting all evidence and discussing the situation), is

reflected by the density pi(θ) = 2(1− θ) (on (0, 1), of course).

Find the Bayes estimator for a. [10 marks]

8. (a) An insurance company has a set of n risks (i = 1, · · · , n) for which it has recorded the number

of claims per year, Yij , for m years (j = 1, · · · ,m).

Suppose the Yij (j = 1, · · · ,m) follows a Poisson distribution with mean exp(βi). Explain

how to find the maximum likelihood estimate for βi.

[3 marks]

(b) State the Central Limit Theorem. [3 marks]

9. An insurance company sells type I car insurance policies for major accidents. Denote N(t) as the

number of type I claims by time t, which follows a Poisson process with rate η. Denote

Xi, i = 1, · · · , N(t) as each claim amounts, which are independent and are exponentially distributed

with mean λ. The random variables Xis are also independent of N(t).

This insurance company also sells type II car insurance policies for minor accidents. Denote Y (t) as

the number of type II claims by time t, which follows a Poisson process with rate δ. For each type II

claim, the claim amount is fixed, having value A.

Let S(t) denote the aggregate claims from the portfolio of the two types of policies, i.e.

S(t) =

∑N(t)

i=1 Xi + AY (t). Show that the mean and variance of S(t) are given by

E(S(t)) = ληt+ Aδt

and

V ar(S(t)) = ηt · (2λ2) + A2δt

[14 marks]

5 MA318-7-AU

10. It is known that the random variable X has probability density function of the form

f(x|θ) =

{

1

θ

e−x/θ, 0 < x <∞

0 otherwise

The sample is X1, X2 of size n = 2.

Suppose that we want to test the hypothesis H0 : θ = 2 against H1 : θ = 4. We choose the critical

region as C = {(x1, x2) : 5 ≤ (x1 + x2)/2 <∞}. Determine the significance level.

[7 marks]

END OF PAPER

6 MA318 – useful distribution densities

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

f(p|α, β) = 1

B(α, β)

pα−1(1− p)β−1, p ∈ [0, 1], α, β > 0

with mean α/(α + β). Here

B(α, β) =

Γ(α)Γ(β)

Γ(α + β)

and Γ(α) has the following property

Γ(α + 1) = αΓ(α).

2 p = (p1, · · · , pk) follows a Dirichlet-distribution, Dirichlet(α1, · · · , αk), then it has pdf

f(p1, · · · , pk|α1, · · · , αk) = Γ(

∑

i αi)

Γ(α1) · · ·Γ(αk)p

α1−1

1 · · · pαk−1k ,

∑

pi = 1, pi ∈ [0, 1], αi > 0

with mean αk/ (

∑

i αi) for pk.

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

f(x|n, p) =

(

n

x

)

px(1− p)n−x, p ∈ [0, 1]

with mean np.

4 θ follows a Normal distribution of mean µ and variance σ2, then it has pdf

f(θ|µ, σ2) = 1√

2piσ2

e−

(θ−µ)2

2σ2

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

f(x|r, 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(x|n,p) = n!

x1! · · ·xk!p

x1

1 · · · pxkk

where

∑

i xi = n, pi ∈ [0, 1] and

∑

i pi = 1.

7 MA318 – useful distribution densities

7 λ follows a Gamma distribution, Gamma(α, β), then it has pdf

f(λ|α, β) = β

α

Γ(α)

λα−1e−βλ, α, β > 0

Note that if α = 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(x|a, b) = 1

b− a, for all x ∈ [a, b];

otherwise f(x|a, b) = 0.

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

p(x|λ) = λ

x

x!

e−λ,