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STAT441-stat441代写

时间：2023-04-02

STAT 441: Homework 3 Due: Monday, 04/10/2023 by 11:59 pm

1. Let {!, ", … , #} be a random sample from the exponential distribution with parameter .

Show that = ∑$ is a sufficient statistic for .

2. Let !, ", … , # be a i.i.d. random sample from a continuous uniform distribution over (, )

where both and are unknown.

(a) Find the Method of Moments (MOM) estimators for and .

(b) Suppose a random sample of 4 numbers are observed:

{1.4, 9.1, 5.8, 8.4}

Estimate the values of and using these sample data.

3. Let {!, ", … , #} be a random sample of size from a chi-square distribution "(), where

the number of degrees of freedom, , is unknown.

(a) Find the method of moment (MOM) estimator of .

(b) Suppose we observed a random sample of 10 values !, ", … , !% from that chi-square

distribution such that 4 $!%$&! = 21.5.

Estimate the value of based on this sample and the estimator you found in part (a).

4. Suppose a discrete random variable has a p.m.f. (|), where ∈ 1,2,3. The following

table provides the family of p.m.f.

(| = 1) (| = 2) (| = 3)

0 1/6 1/4 1/5

1 1/4 1/4 1/5

2 0 1/4 1/5

3 1/2 1/8 1/5

4 1/12 1/8 1/5

(a) Suppose = 2 is observed. What is the Maximum Likelihood Estimator (MLE) of ?

(b) Suppose a random sample of size 3, which is (!, ", ') = (1,3,4) is observed. What is

the MLE of ?

STAT 441: Homework 3 Due: Monday, 04/10/2023 by 11:59 pm

5. Let {!, ", … , #} be a random sample from a gamma distribution with parameters = 2

and (unknown).

a) Find the log-likelihood function ln (!, … , #|).

b) Find the maximum likelihood estimator (MLE) of .

6. Refer to Question 5.

a) Show that the gamma distribution with parameters = 2 and (unknown) is an

exponential family distribution.

b) Denote the variance of (2, ) distribution by . Find the MLE of . (Hint: is a

function of .)

7. The past records of a supermarket show that its customers spend on average $65 per visit at

this store. Recently the management of the store initiated a promotional campaign according

to which each customer receives points based on the total money spent at the store and these

points can be used to buy products at the store. The management expects that as a result of

this campaign, the customers would be encouraged to spend more money at the store. To

check whether this is true, the manager of the store took a random sample of 12 customers

who visited the store. The following data give the amount (in dollars) spent by those

customers.

88, 69, 141, 28, 106, 45, 32, 51, 78, 54, 110, 83

Assume that the amount spent by all customers at this supermarket has a normal distribution

with mean and variance ". Answer the following questions.

(a) Compute the sample mean and the sample standard deviation of the above sample.

(b) What are the unbiased estimators of and "?

(c) What are the estimates of and " based on this sample?

(d) Construct a 95% confidence interval for . Interpret the interval in the context of the

study.

1. Let {!, ", … , #} be a random sample from the exponential distribution with parameter .

Show that = ∑$ is a sufficient statistic for .

2. Let !, ", … , # be a i.i.d. random sample from a continuous uniform distribution over (, )

where both and are unknown.

(a) Find the Method of Moments (MOM) estimators for and .

(b) Suppose a random sample of 4 numbers are observed:

{1.4, 9.1, 5.8, 8.4}

Estimate the values of and using these sample data.

3. Let {!, ", … , #} be a random sample of size from a chi-square distribution "(), where

the number of degrees of freedom, , is unknown.

(a) Find the method of moment (MOM) estimator of .

(b) Suppose we observed a random sample of 10 values !, ", … , !% from that chi-square

distribution such that 4 $!%$&! = 21.5.

Estimate the value of based on this sample and the estimator you found in part (a).

4. Suppose a discrete random variable has a p.m.f. (|), where ∈ 1,2,3. The following

table provides the family of p.m.f.

(| = 1) (| = 2) (| = 3)

0 1/6 1/4 1/5

1 1/4 1/4 1/5

2 0 1/4 1/5

3 1/2 1/8 1/5

4 1/12 1/8 1/5

(a) Suppose = 2 is observed. What is the Maximum Likelihood Estimator (MLE) of ?

(b) Suppose a random sample of size 3, which is (!, ", ') = (1,3,4) is observed. What is

the MLE of ?

STAT 441: Homework 3 Due: Monday, 04/10/2023 by 11:59 pm

5. Let {!, ", … , #} be a random sample from a gamma distribution with parameters = 2

and (unknown).

a) Find the log-likelihood function ln (!, … , #|).

b) Find the maximum likelihood estimator (MLE) of .

6. Refer to Question 5.

a) Show that the gamma distribution with parameters = 2 and (unknown) is an

exponential family distribution.

b) Denote the variance of (2, ) distribution by . Find the MLE of . (Hint: is a

function of .)

7. The past records of a supermarket show that its customers spend on average $65 per visit at

this store. Recently the management of the store initiated a promotional campaign according

to which each customer receives points based on the total money spent at the store and these

points can be used to buy products at the store. The management expects that as a result of

this campaign, the customers would be encouraged to spend more money at the store. To

check whether this is true, the manager of the store took a random sample of 12 customers

who visited the store. The following data give the amount (in dollars) spent by those

customers.

88, 69, 141, 28, 106, 45, 32, 51, 78, 54, 110, 83

Assume that the amount spent by all customers at this supermarket has a normal distribution

with mean and variance ". Answer the following questions.

(a) Compute the sample mean and the sample standard deviation of the above sample.

(b) What are the unbiased estimators of and "?

(c) What are the estimates of and " based on this sample?

(d) Construct a 95% confidence interval for . Interpret the interval in the context of the

study.