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.
The value of the limit
is equal to
Options-
Answer:
If the series converges absolutely, then which of the following series diverges?
Options-
Answer:
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..
Answer:
Solution:
Let be a sequence of independent and identically distributed random variables with probability density function
Then. the value of the limit
Options -
Answer:
Let be a function defined by
Then, which of the following statements is TRUE?
Options -
Answer: is both one-one and onto
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 -
Answer:
Let be a random variable with probability density function
Then, which of the following statements is FALSE?
Options -
Answer:
The value of the limit
.
is equal to
Options-
Answer:
Solution:
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.
Answer:
The value of the limit
Options -
Answer:
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.
Answer:
Let,
Then, which of the following statements is TRUE?
Options -
1.
2.
Answer:
Let and define recursively
Then, which of the following statements is TRUE?
Options-
Answer: is monotone decreasing, and
Let and
be three events such that
and
Then. which of the following statements is FALSE?
Answer:
Let and
be four events such that
Answer:
Let be a random variable having the probability density function
Define , where
denotes the largest integer not exceeding
. Then,
is equal to
Options -
1.
Answer:
Let be a continuous random variable with distribution function
for some real constant . Then,
is equal to
Options -
1.1
2 .
Answer:
Let be a random sample from
where
is unknown. Let
and
Then, which of the following statements is TRUE?
Options -
Answer: is an
of
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 -
Answer: 0.81
Solution:
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 -
Answer:
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 -
Answer: exists and
is continuous at (0,0)
Let be a random vector with joint moment generating function
Let . Then.
is equal to
Options -
1.3
2.4
3.5
4.6
Answer: 5
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]
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 -
Answer:
For , consider the system of linear equations
in the unknowns and
. Then. which of the following statements is
?
Options -
Answer: The given system has a unique solution for
Let and
be independent
random variables and
Then, which of the
following expectations is finite?
Options -
Answer:
Let be a sequence of independent and identically distributed
random variables.
Then,
Options -
Answer:
Let be a continuous random variable having the moment generating function
Let and
.
Then, the value of is equal to
Options-
Answer:
Let be a random sample from Poisson
where
is unknown and
let
Then, the uniformly minimum variance unbiased estimator of
Options -
Answer:
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
Answer:
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.
The value of the limit
is equal to
Options-
Answer:
If the series converges absolutely, then which of the following series diverges?
Options-
Answer:
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..
Answer:
Solution:
Let be a sequence of independent and identically distributed random variables with probability density function
Then. the value of the limit
Options -
Answer:
Let be a function defined by
Then, which of the following statements is TRUE?
Options -
Answer: is both one-one and onto
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 -
Answer:
Let be a random variable with probability density function
Then, which of the following statements is FALSE?
Options -
Answer:
The value of the limit
.
is equal to
Options-
Answer:
Solution:
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.
Answer:
The value of the limit
Options -
Answer:
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.
Answer:
Let,
Then, which of the following statements is TRUE?
Options -
1.
2.
Answer:
Let and define recursively
Then, which of the following statements is TRUE?
Options-
Answer: is monotone decreasing, and
Let and
be three events such that
and
Then. which of the following statements is FALSE?
Answer:
Let and
be four events such that
Answer:
Let be a random variable having the probability density function
Define , where
denotes the largest integer not exceeding
. Then,
is equal to
Options -
1.
Answer:
Let be a continuous random variable with distribution function
for some real constant . Then,
is equal to
Options -
1.1
2 .
Answer:
Let be a random sample from
where
is unknown. Let
and
Then, which of the following statements is TRUE?
Options -
Answer: is an
of
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 -
Answer: 0.81
Solution:
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 -
Answer:
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 -
Answer: exists and
is continuous at (0,0)
Let be a random vector with joint moment generating function
Let . Then.
is equal to
Options -
1.3
2.4
3.5
4.6
Answer: 5
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]
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 -
Answer:
For , consider the system of linear equations
in the unknowns and
. Then. which of the following statements is
?
Options -
Answer: The given system has a unique solution for
Let and
be independent
random variables and
Then, which of the
following expectations is finite?
Options -
Answer:
Let be a sequence of independent and identically distributed
random variables.
Then,
Options -
Answer:
Let be a continuous random variable having the moment generating function
Let and
.
Then, the value of is equal to
Options-
Answer:
Let be a random sample from Poisson
where
is unknown and
let
Then, the uniformly minimum variance unbiased estimator of
Options -
Answer:
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
Answer:
Please claim problem 11 to official answer key