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IIT JAM MS 2021 Question Paper | Set A | Problems & Solutions

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

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

Problem 1

The value of the limit

is equal to

Options-

Problem 2

If the series converges absolutely, then which of the following series diverges?

Options-

Problem 3

Let be a random variable and let . If is the correlation coefficient between the random variables and , then is equal to

Options-

1.;
2.;
3.;
4..

Solution:

Problem 4

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

f(x)={1,0, if 0<x<1 otherwise

Then. the value of the limit

limn→∞P(1ni=1nlnXi≤1+1n)
is equal to

Options -

1. ;
2. ;
3. ;
4. .

Problem 5

Let be a function defined by

Then, which of the following statements is TRUE?

Options -

1. is onto but NOT one-one
2. is one-one but NOT onto
3. is both one-one and onto
4. is neither one-one nor onto

Answer: is both one-one and onto

Problem 6

There are three urns, labeled. Urn , Urn and Urn . Urn contains white balls and black balls, Urn contains white ball and black balls and Urn contains white balls and black ball. Consider two coins with probability of obtaining head in their single trials as and The two coins are tossed independently once, and an urn is selected according to the following scheme:
Urn is selected if heads are obtained: Urn is selected if tails are obtained; otherwise Urn is
selected. A ball is then drawn at random from the selected urn. Then
Urn 1 is selected the ball drawn is white is equal to

Options -

Problem 7

Let be a random variable with probability density function

Then, which of the following statements is FALSE?

Options -

Problem 8

The value of the limit

.

is equal to

Options-

Solution:

Problem 9

Let be a real matrix. Let and be the eigenvectors of corresponding to three distinct eigenvalues of . where is a real number. Then. which of the following is NOT a possible value of ?

Options-

1.
2 .
3.
4.

Problem 10

The value of the limit

is equal to

Options -

1. 0
2. 1

Problem 11

Consider a sequence of independent Bernoulli trials with probability of success in each trial as . The probability that three successes occur before four failures is equal to

Options -

1.
2.
3.
4.

Problem 12

Let,

Then, which of the following statements is TRUE?

Options -

1.
2.

IIT JAM 2021 - Problem 13

Let and define recursively

Then, which of the following statements is TRUE?

Options-

1. is monotone decreasing, and
2. is decreasing, and
3. is non-monotone, and
4. is monotone increasing, and

Problem 14

Let and be three events such that and
Then. which of the following statements is FALSE?

Problem 15

Let and be four events such that

Then. is equal to

Problem 16

Let be a random variable having the probability density function

Define , where denotes the largest integer not exceeding . Then, is equal to

Options -

1.

Problem 17

Let be a continuous random variable with distribution function

for some real constant . Then, is equal to

Options -

1.1
2 .

1. 0

Problem 18

Let be a random sample from where is unknown. Let and Then, which of the following statements is TRUE?

Options -

1. is an MLE of
2. is the unique of
3. MLE of does NOT exist
4. is an of

Problem 19

Consider the problem of testing against based on a sample of size 1 , where

and .

Then, the probability of Type II error of the most powerful test of size is equal to

Options -

1. 0.1
2. 1
3. 0.91
4. 0.81

Solution:

Problem 20

Let and be random variables having chi-square distributions with 6 and 3 degrees of freedom, respectively. Then, which of the following statements is TRUE?

Options -

Problem 21

Let be a function defined by

.

Let and denote the first order partial derivatives of with respect to and ,
respectively, at the point . Then, which of the following statements is FALSE?

Options -

1. is NOT differentiable at (0,0)
2. exists and is continuous at (0,0)
3. exists and is bounded at every
4. exists and is bounded at every

Answer: exists and is continuous at (0,0)

Problem 22

Let be a random vector with joint moment generating function

Let . Then. is equal to

Options -

1.3

2.4

3.5

4.6

Problem 23

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

where is unknown. Let be fixed and let be the power of the most powerful test of size for testing against .
Consider the critical region

where for any is a fixed point such that Then, the
critical region corresponds to the

Options-

1.most powerful test of size for testing against
2.most powerful test of size for testing against
3.most powerful test of size for testing against
4.most powerful test of size for testing against

Answer: most powerful test of size for testing against [No Option]

Problem 24

Consider three coins having probabilities of obtaining head in a single trial as and , respectively, A player selects one of these three coins at random (each coin is equally likely to be selected). If the player tosses the selected coin five times independently, then the probability of obtaining two tails in five tosses is equal to

Options -

Problem 25

For , consider the system of linear equations

in the unknowns and . Then. which of the following statements is ?

Options -

1. The given system has a unique solution for
2. The given system has a unique solution for
3. The given system has infinitely many solutions for
4. The given system has infinitely many solutions for

Answer: The given system has a unique solution for

Problem 26

Let and be independent random variables and Then, which of the
following expectations is finite?

Options -

Problem 27

Let be a sequence of independent and identically distributed random variables.
Then,

is equal to

Options -

1. 0

Problem 28

Let be a continuous random variable having the moment generating function

Let and .
Then, the value of is equal to

Options-

Problem 29

Let be a random sample from Poisson where is unknown and
let

Then, the uniformly minimum variance unbiased estimator of

Options -

1. is
2. is
3. does NOT exist
4. is

Problem 30

Let be a sequence of real numbers such that , for all . Then, which of the following conditions imply the divergence of

Options -

1 converges, where and for all

1. converges
2. is non-increasing

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

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

Problem 1

The value of the limit

is equal to

Options-

Problem 2

If the series converges absolutely, then which of the following series diverges?

Options-

Problem 3

Let be a random variable and let . If is the correlation coefficient between the random variables and , then is equal to

Options-

1.;
2.;
3.;
4..

Solution:

Problem 4

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

f(x)={1,0, if 0<x<1 otherwise

Then. the value of the limit

limn→∞P(1ni=1nlnXi≤1+1n)
is equal to

Options -

1. ;
2. ;
3. ;
4. .

Problem 5

Let be a function defined by

Then, which of the following statements is TRUE?

Options -

1. is onto but NOT one-one
2. is one-one but NOT onto
3. is both one-one and onto
4. is neither one-one nor onto

Answer: is both one-one and onto

Problem 6

There are three urns, labeled. Urn , Urn and Urn . Urn contains white balls and black balls, Urn contains white ball and black balls and Urn contains white balls and black ball. Consider two coins with probability of obtaining head in their single trials as and The two coins are tossed independently once, and an urn is selected according to the following scheme:
Urn is selected if heads are obtained: Urn is selected if tails are obtained; otherwise Urn is
selected. A ball is then drawn at random from the selected urn. Then
Urn 1 is selected the ball drawn is white is equal to

Options -

Problem 7

Let be a random variable with probability density function

Then, which of the following statements is FALSE?

Options -

Problem 8

The value of the limit

.

is equal to

Options-

Solution:

Problem 9

Let be a real matrix. Let and be the eigenvectors of corresponding to three distinct eigenvalues of . where is a real number. Then. which of the following is NOT a possible value of ?

Options-

1.
2 .
3.
4.

Problem 10

The value of the limit

is equal to

Options -

1. 0
2. 1

Problem 11

Consider a sequence of independent Bernoulli trials with probability of success in each trial as . The probability that three successes occur before four failures is equal to

Options -

1.
2.
3.
4.

Problem 12

Let,

Then, which of the following statements is TRUE?

Options -

1.
2.

IIT JAM 2021 - Problem 13

Let and define recursively

Then, which of the following statements is TRUE?

Options-

1. is monotone decreasing, and
2. is decreasing, and
3. is non-monotone, and
4. is monotone increasing, and

Problem 14

Let and be three events such that and
Then. which of the following statements is FALSE?

Problem 15

Let and be four events such that

Then. is equal to

Problem 16

Let be a random variable having the probability density function

Define , where denotes the largest integer not exceeding . Then, is equal to

Options -

1.

Problem 17

Let be a continuous random variable with distribution function

for some real constant . Then, is equal to

Options -

1.1
2 .

1. 0

Problem 18

Let be a random sample from where is unknown. Let and Then, which of the following statements is TRUE?

Options -

1. is an MLE of
2. is the unique of
3. MLE of does NOT exist
4. is an of

Problem 19

Consider the problem of testing against based on a sample of size 1 , where

and .

Then, the probability of Type II error of the most powerful test of size is equal to

Options -

1. 0.1
2. 1
3. 0.91
4. 0.81

Solution:

Problem 20

Let and be random variables having chi-square distributions with 6 and 3 degrees of freedom, respectively. Then, which of the following statements is TRUE?

Options -

Problem 21

Let be a function defined by

.

Let and denote the first order partial derivatives of with respect to and ,
respectively, at the point . Then, which of the following statements is FALSE?

Options -

1. is NOT differentiable at (0,0)
2. exists and is continuous at (0,0)
3. exists and is bounded at every
4. exists and is bounded at every

Answer: exists and is continuous at (0,0)

Problem 22

Let be a random vector with joint moment generating function

Let . Then. is equal to

Options -

1.3

2.4

3.5

4.6

Problem 23

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

where is unknown. Let be fixed and let be the power of the most powerful test of size for testing against .
Consider the critical region

where for any is a fixed point such that Then, the
critical region corresponds to the

Options-

1.most powerful test of size for testing against
2.most powerful test of size for testing against
3.most powerful test of size for testing against
4.most powerful test of size for testing against

Answer: most powerful test of size for testing against [No Option]

Problem 24

Consider three coins having probabilities of obtaining head in a single trial as and , respectively, A player selects one of these three coins at random (each coin is equally likely to be selected). If the player tosses the selected coin five times independently, then the probability of obtaining two tails in five tosses is equal to

Options -

Problem 25

For , consider the system of linear equations

in the unknowns and . Then. which of the following statements is ?

Options -

1. The given system has a unique solution for
2. The given system has a unique solution for
3. The given system has infinitely many solutions for
4. The given system has infinitely many solutions for

Answer: The given system has a unique solution for

Problem 26

Let and be independent random variables and Then, which of the
following expectations is finite?

Options -

Problem 27

Let be a sequence of independent and identically distributed random variables.
Then,

is equal to

Options -

1. 0

Problem 28

Let be a continuous random variable having the moment generating function

Let and .
Then, the value of is equal to

Options-

Problem 29

Let be a random sample from Poisson where is unknown and
let

Then, the uniformly minimum variance unbiased estimator of

Options -

1. is
2. is
3. does NOT exist
4. is

Problem 30

Let be a sequence of real numbers such that , for all . Then, which of the following conditions imply the divergence of

Options -

1 converges, where and for all

1. converges
2. is non-increasing

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