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统计代写-S203031

时间：2021-01-24

Prof. Elvezio Ronchetti Research Center for Statistics and GSEM

Ass : Julien Bodelet University of Geneva

S203031 Probability and Statistics II Fall 2019

EXAM January 17, 2020

— The exam lasts 3h.

— You may answer in English or in French.

— You are not allowed to leave the room during the exam.

— Open book exam.

— Justify your answers and do not forget to write your name on your copy.

Good luck !

Exercise 1 ( 11 points)

Let X1, X2, . . . , Xn be n i.i.d random variables representing the prices set by n competitors for a

given product and assume that Xi follows an exponential distribution with parameter λ, i.e. with density

fX(x) =

{

λe−λx if x > 0

0 otherwise ,

where λ is a positive unknown parameter. We want to investigate the statistical properties of the lowest

price in the market for the product, Y = min(X1, X2, ..., Xn).

1. Compute the cumulative distribution function FY (y) of Y .

2. Show that the density of Y is exponential with parameter nλ.

3. Compute the expected value and the interquartile range of Y .

4. We simulate M = 106 samples of size n = 10 of prices issued from an exponential distribution

with parameter λ = 1. For each sample we take the lowest price. What is the approximate average

value of the M lowest prices? What is the approximate percentage among the M lowest prices

which are larger than 0.1386? Justify your answers.

Exercise 2 ( 7 points)

LetX1, X2 be two independent random variables representing the number of insurance claims during

a month for two groups of drivers. We assume that Xi follows a Poisson distribution with parameter

λi, i = 1, 2.

1. Show that for every n ≥ 1, the conditional distribution of X1, given X1 + X2 = n, is binomial

and give the parameters of this binomial distribution.

2. Assume that the expected number of claims in each group is the same, i.e. E[X1] = E[X2]. If

the total number of claims for the two groups is 150, what is the probability that the number of

claims for the first group is at least 90?

Hint : Use the normal approximation.

Exercise 3 ( 12 points)

Let X1, X2, . . . , Xn be i.i.d random variables from a continuous distribution with density

fα(x) = α(α + 1)x

α−1(1− x) ,

where α > 0 and 0 < x < 1.

1. Show that µ = E[Xi] = αα+2 .

2. Show that αˆn = g(X¯n) = 2X¯n/(1 − X¯n) is the Method of Moments estimator of α. Is this

estimator consistent for α?

3. Let σ2 = var[X] = 2α

(α+2)2(α+3)

. Give the distribution of

√

n(X¯n − µ), when n→∞.

4. Show that

√

n(g(X¯n)− g(µ))→ N

(

0,

(

dg(µ)

dµ

)2

σ2

)

,

when n→∞.

Hint : Obtain an approximation of g(X¯n) by means of a first order Taylor approximation around

µ.

5. Using point 4., show that the distribution of

√

n(αˆn − α) isN (0, V (α)), when n→∞ and give

the function V (α).

6. We would like to use the estimator αˆn for testing. Write the the corresponding Studentized sta-

tistic and using point 5. give its distribution, when n→∞.

Ass : Julien Bodelet University of Geneva

S203031 Probability and Statistics II Fall 2019

EXAM January 17, 2020

— The exam lasts 3h.

— You may answer in English or in French.

— You are not allowed to leave the room during the exam.

— Open book exam.

— Justify your answers and do not forget to write your name on your copy.

Good luck !

Exercise 1 ( 11 points)

Let X1, X2, . . . , Xn be n i.i.d random variables representing the prices set by n competitors for a

given product and assume that Xi follows an exponential distribution with parameter λ, i.e. with density

fX(x) =

{

λe−λx if x > 0

0 otherwise ,

where λ is a positive unknown parameter. We want to investigate the statistical properties of the lowest

price in the market for the product, Y = min(X1, X2, ..., Xn).

1. Compute the cumulative distribution function FY (y) of Y .

2. Show that the density of Y is exponential with parameter nλ.

3. Compute the expected value and the interquartile range of Y .

4. We simulate M = 106 samples of size n = 10 of prices issued from an exponential distribution

with parameter λ = 1. For each sample we take the lowest price. What is the approximate average

value of the M lowest prices? What is the approximate percentage among the M lowest prices

which are larger than 0.1386? Justify your answers.

Exercise 2 ( 7 points)

LetX1, X2 be two independent random variables representing the number of insurance claims during

a month for two groups of drivers. We assume that Xi follows a Poisson distribution with parameter

λi, i = 1, 2.

1. Show that for every n ≥ 1, the conditional distribution of X1, given X1 + X2 = n, is binomial

and give the parameters of this binomial distribution.

2. Assume that the expected number of claims in each group is the same, i.e. E[X1] = E[X2]. If

the total number of claims for the two groups is 150, what is the probability that the number of

claims for the first group is at least 90?

Hint : Use the normal approximation.

Exercise 3 ( 12 points)

Let X1, X2, . . . , Xn be i.i.d random variables from a continuous distribution with density

fα(x) = α(α + 1)x

α−1(1− x) ,

where α > 0 and 0 < x < 1.

1. Show that µ = E[Xi] = αα+2 .

2. Show that αˆn = g(X¯n) = 2X¯n/(1 − X¯n) is the Method of Moments estimator of α. Is this

estimator consistent for α?

3. Let σ2 = var[X] = 2α

(α+2)2(α+3)

. Give the distribution of

√

n(X¯n − µ), when n→∞.

4. Show that

√

n(g(X¯n)− g(µ))→ N

(

0,

(

dg(µ)

dµ

)2

σ2

)

,

when n→∞.

Hint : Obtain an approximation of g(X¯n) by means of a first order Taylor approximation around

µ.

5. Using point 4., show that the distribution of

√

n(αˆn − α) isN (0, V (α)), when n→∞ and give

the function V (α).

6. We would like to use the estimator αˆn for testing. Write the the corresponding Studentized sta-

tistic and using point 5. give its distribution, when n→∞.