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# IIT JAM MS 2021 Question Paper | Set B | Problems & Solutions

This post discusses the solutions to the problems from IIT JAM Mathematical Statistics (MS) 2021 Question Paper - Set B. You can find solutions in video or written form.

Note: This post is getting updated. Stay tuned for solutions, videos, and more.

## IIT JAM Mathematical Statistics (MS) 2021 Problems & Solutions (Set B)

### Problem 1

A sample of size is drawn randomly (without replacement) from an urn couraining balls, of which are red balls and are black balls. Let denote the number of red balls in the selected sample. If and then which of the following statements is/are TR UE?

Options -

### Problem 2

Let be a random sample from a distribution with probability density function

where is unknown.
If , then which of the following statements is/are TRUE?

Options -

1. is the unique uniformly minimum variance unbiased estimator of
2. is the unique uniformly minimum variance unbiased estimator of
3. is the unique uniformly minimum variance unbiased estimator of

1. is the MLE of

is the unique uniformly minimum variance unbiased estimator of
is the MLE of

### Problem 3

Consider the linear system , where is an matrix, is an vector of unknowns
and is an vector. Further, suppose there exists an vector such that the linear system has No solution. Then, which of the following statements is/are necessarily TRUE?

Options -

1.If and is the first column of , then the linear system has a unique solution
2.If then the linear system has a solution other than

1. If then

.

### Problem 4

Let be independent and identically distributed random variables with probability density function

Then, which of the following random variables has/have finite expectation?

Options -

### Problem 5

Let be a random sample from where is unknown. Consider the problem of testing against Let denote the power function of the likelihood ratio test of size for testing against . Then. which of the following statements is/are TRUE?

Options -

1.The critical region of the likelihood test of size is

where is a fixed point such that

1. for all
2. The critical region of the likelihood test of size is
{(x1,x2,…,xn)Rn:nni=1xin>τα/2}

where is a fixed point such that
3. for all

### Problem 6

Let be a random sample from a distribution with probability density function

where is unknown. If and then which
of the following statements is/are TRUE?

Options -

1. is a complete and sufficient statistic for

1. is an of
2. is jointly sufficient for
3. Distribution of does NOT depend on

is a complete and sufficient statistic for
is jointly sufficient for
Distribution of does NOT depend on

### Problem 7

Let be a random sample from a distribution with probability density function

where is unknown. Then, which of the following statements is/are TRUE?

Options -

1 .There does NOT exist any unbiased estimator of which attains the Cramer-Rao lower bound
2.Cramer-Rao lower bound, based on for the estimand is
3 .Cramer-Rao lower bound. based on for the estimand is
4 .There exists an unbiased estimator of which attains the Cramer-Rao lower bound

Cramer-Rao lower bound. based on for the estimand is
There exists an unbiased estimator of which attains the Cramer-Rao lower bound

### Problem 8

Let be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?

Options-

1. is continuous
2. is bounded on (0,1)
3. If then has a solution in (0,1)
4. is bounded on [8,10]

If then has a solution in (0,1)
is bounded on [8,10]

### Problem 9

Let be a real matrix such that and the sum of the entries in each row of is . Then which of the following statements is/are necessarily TRUE?

Options -

1. The characteristic polynomial, of has as a factor
1. is an invertible matrix
2. cannot be an orthogonal matrix
3. The set has at least two elements is a column vector)

The characteristic polynomial, of has as a factor

### Problem 10

Consider the function

If , then which of the following statements is/are TRUE?

Options -

1. The maximum value of on is
2. The maximum value of on is
3. The minimum value of on is
4. The minimum value of on is

The maximum value of on is
The minimum value of on is

This post discusses the solutions to the problems from IIT JAM Mathematical Statistics (MS) 2021 Question Paper - Set B. You can find solutions in video or written form.

Note: This post is getting updated. Stay tuned for solutions, videos, and more.

## IIT JAM Mathematical Statistics (MS) 2021 Problems & Solutions (Set B)

### Problem 1

A sample of size is drawn randomly (without replacement) from an urn couraining balls, of which are red balls and are black balls. Let denote the number of red balls in the selected sample. If and then which of the following statements is/are TR UE?

Options -

### Problem 2

Let be a random sample from a distribution with probability density function

where is unknown.
If , then which of the following statements is/are TRUE?

Options -

1. is the unique uniformly minimum variance unbiased estimator of
2. is the unique uniformly minimum variance unbiased estimator of
3. is the unique uniformly minimum variance unbiased estimator of

1. is the MLE of

is the unique uniformly minimum variance unbiased estimator of
is the MLE of

### Problem 3

Consider the linear system , where is an matrix, is an vector of unknowns
and is an vector. Further, suppose there exists an vector such that the linear system has No solution. Then, which of the following statements is/are necessarily TRUE?

Options -

1.If and is the first column of , then the linear system has a unique solution
2.If then the linear system has a solution other than

1. If then

.

### Problem 4

Let be independent and identically distributed random variables with probability density function

Then, which of the following random variables has/have finite expectation?

Options -

### Problem 5

Let be a random sample from where is unknown. Consider the problem of testing against Let denote the power function of the likelihood ratio test of size for testing against . Then. which of the following statements is/are TRUE?

Options -

1.The critical region of the likelihood test of size is

where is a fixed point such that

1. for all
2. The critical region of the likelihood test of size is
{(x1,x2,…,xn)Rn:nni=1xin>τα/2}

where is a fixed point such that
3. for all

### Problem 6

Let be a random sample from a distribution with probability density function

where is unknown. If and then which
of the following statements is/are TRUE?

Options -

1. is a complete and sufficient statistic for

1. is an of
2. is jointly sufficient for
3. Distribution of does NOT depend on

is a complete and sufficient statistic for
is jointly sufficient for
Distribution of does NOT depend on

### Problem 7

Let be a random sample from a distribution with probability density function

where is unknown. Then, which of the following statements is/are TRUE?

Options -

1 .There does NOT exist any unbiased estimator of which attains the Cramer-Rao lower bound
2.Cramer-Rao lower bound, based on for the estimand is
3 .Cramer-Rao lower bound. based on for the estimand is
4 .There exists an unbiased estimator of which attains the Cramer-Rao lower bound

Cramer-Rao lower bound. based on for the estimand is
There exists an unbiased estimator of which attains the Cramer-Rao lower bound

### Problem 8

Let be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?

Options-

1. is continuous
2. is bounded on (0,1)
3. If then has a solution in (0,1)
4. is bounded on [8,10]

If then has a solution in (0,1)
is bounded on [8,10]

### Problem 9

Let be a real matrix such that and the sum of the entries in each row of is . Then which of the following statements is/are necessarily TRUE?

Options -

1. The characteristic polynomial, of has as a factor
1. is an invertible matrix
2. cannot be an orthogonal matrix
3. The set has at least two elements is a column vector)

The characteristic polynomial, of has as a factor

### Problem 10

Consider the function

If , then which of the following statements is/are TRUE?

Options -

1. The maximum value of on is
2. The maximum value of on is
3. The minimum value of on is
4. The minimum value of on is

The maximum value of on is
The minimum value of on is

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