This post discusses the solutions to the problems from IIT JAM Mathematical Statistics (MS) 2021 Question Paper - Set C. 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 C)

Problem 1

Let $f_{0}$ and $f_{1}$ be the probability mass functions given by

Consider the problem of testing the mull hypothesis $H_{0}: X \sim f_{0}$ a gainst $H_{1}: X \sim f_{1}$ based on a single
sample $X .$ If $\alpha$ and $\beta$, respectively, denote the size and power of the test with critical region
${x \in \mathbb{R}: x>3},$ then $10(\alpha+\beta)$ is equal to ______________________

Answer: $13$

Problem 2

Let,

$$
\alpha=\lim _{n \rightarrow \infty} \sum{m=n^{2}}^{2 n^{2}} \frac{1}{\sqrt{5 n^{4}+n^{3}+m}}
$$

Then, $10 \sqrt{5} \alpha$ is equal to _________

Answer: 10

Problem 3

Let $\alpha, \beta$ and $\gamma$ be the eigenvalues of $M=\left[\begin{array}{ccc}0 & 1 & 0 \\ 1 & 3 & 3 \\ -1 & 2 & 2\end{array}\right] .$ If $y=1$ and $\alpha>\beta,$ then the value of
$2 \alpha+3 \beta$ is ___________________________________

Answer: $7$

Problem 4

Let $S=\{(x, y) \in \mathbb{R}^{2}: 2 \leq x \leq y \leq 4\}$. Then, the value of the integral

$$
\iint_{S} \frac{1}{4-x} d x d y
$$

is _______

Answer: 2

Problem 5

Let $M=\left(\begin{array}{cc}5 & -6 \ 3 & -4\end{array}\right)$ be a $2 \times 2$ matrix. If $\alpha=det \left(M^{4}-6 I_{2}\right),$ then the value of $\alpha^{2}$ is ________

Answer: 2500

Problem 6

Let $X$ be a random variable with moment generating function

$$
M_{X}(t)=\frac{1}{12}+\frac{1}{6} e^{t}+\frac{1}{3} e^{2 t}+\frac{1}{4} e^{-t}+\frac{1}{6} e^{-2 t}, t \in \mathbb{R}
$$

Then, $8 E(X)$ is equal to _______

Answer: 2

Problem 7

Let $5,10,4,15,6$ be an observed random sample of size 5 from a distribution with probability density function

$$
f(x ; \theta)=\begin{cases}
e^{-(x-\theta)}, x \geq \theta \\
0, \text { otherwise }
\end{cases}.
$$

$\theta \in(-\infty, 3]$ is unknown. Then, the maximum likelihood estimate of $\theta$ based on the observed sample is equal to ________

Answer: 3

Problem 8

Let $X$ be a random variable having the probability density function

$$
f(x)=\frac{1}{8 \sqrt{2 \pi}}\left(2 e^{-\frac{x^{2}}{2}}+3 e^{-\frac{x^{2}}{8}}\right), \quad-\infty<x<\infty .
$$

Then, $4 E\left(X^{4}\right)$ is equal to _____

Answer: 147

Problem 9

Let $\beta$ denote the length of the curve $y=\ln (\sec x)$ from $x=0$ to $x=\frac{\pi}{4}$. Then, the value of $3 \sqrt{2}\left(e^{\beta}-1\right)$ is equal to _____

Answer: $6$

Problem 10

Let $A=\{(x, y, z) \in \mathbb{R}^{3}: 0 \leq x \leq y \leq z \leq 1\}$. Let $\alpha$ be the value of the integral

$$
\iiint_{A} x y z d x d y d z
$$

Then, $384 \alpha$ is equal to _______

Answer: $8$

Problem 11

Let,

$$
a_{n}=\sum_{k=2}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) \frac{2^{k}(n-2)^{n-k}}{n^{n}}, \quad n=2,3, \ldots
$$

Then, $e^{2} \lim _{n \rightarrow \infty}\left(1-a{n}\right)$ is equal to ____

Answer: 3

Problem 12

Let $E_{1}, E_{2}, E_{3}$ and $E_{4}$ be four independent events such that $P\left(E_{1}\right)=\frac{1}{2}, P\left(E_{2}\right)=\frac{1}{3}, P\left(E_{3}\right)=\frac{1}{4}$ and $P\left(E_{4}\right)=\frac{1}{5} .$ Let $p$ be the probability that at most two events among $E_{1}, E_{2}, E_{3}$ and $E_{4}$ occur. Then, $240 p$ is equal to ____

Answer: 218

Problem 13

The number of real roots of the polynomial

$$
f(x)=x^{11}-13 x+5
$$

is ____

Answer:$3$

Problem 14

Let $S \subseteq \mathbb{R}^{2}$ be the region bounded by the parallelogram with vertices at the points (1,0),(3,2) ,
(3,5) and $(1,3) .$ Then. the value of the integral $\iint_{S}(x+2 y) d x d y$ is equal to ___

Answer: 42

Problem 15

Let $\alpha=\lim _{n \rightarrow \infty}\left(1+n \sin \frac{3}{n^{2}}\right)^{2 n}$. Then, $\ln \alpha$ is equal to ____

Answer: 6

Problem 16

Let $A=\{(x, y) \in \mathbb{R}^{2}: x^{2}-\frac{1}{2 \sqrt{\pi}}<y<x^{2}+\frac{1}{2 \sqrt{\pi}}\}$ and let the joint probability density function
of $(X, Y)$ be

$$
f(x, y)=\begin{cases}
e^{-(x-1)^{2}}, & (x, y) \in A \\
0, \text { otherwise }
\end{cases}.
$$

Then, the covariance between the random variables $X$ and $Y$ is equal to ____

Answer: 1

Problem 17

Let $\phi:(-1,1) \rightarrow \mathbb{R}$ be defined by

$$
\phi(x)=\int_{x^{7}}^{x^{4}} \frac{1}{1+t^{3}} d t
$$

If $\alpha=\lim _{x \rightarrow 0} \frac{\phi(x)}{e^{2 x^{4}-1}},$ then $42 \alpha$ is equal to ____

Answer: 21

Problem 18

Let $S=\{(x, y) \in \mathbb{R}^{2} ; 0 \leq x \leq \pi, \min {\sin x, \cos x} \leq y \leq \max {\sin x, \cos x}\}$.
If $\alpha$ is the area of $S$, then the value of $2 \sqrt{2} \alpha$ is equal to ____

Answer: 8

Problem 19

Let the random vector $(X, Y)$ have the joint probability mass function

$f(x, y)=\begin{cases}{10 \choose x}{5 \choose y}(\frac{1}{4})^{x-y+5}(\frac{3}{4})^{y-x+10}, x=0,1, \ldots, 10 ; y=0,1, \ldots, 5 \\ 0, \text { otherwise }\end{cases}$.

Let $Z=Y-X+10 .$ If $\alpha=E(Z)$ and $\beta=Var(Z),$ then $8 \alpha+48 \beta$ is equal to ____

Answer: 225

Problem 20

Let $X_{1}$ and $X_{2}$ be independent $N(0,1)$ random variables. Define

$$
sgn(u)=\begin{cases}
-1, \text { if } u<0 \\ 0, \text { if } u=0 \\ 1, \text { if } u>0
\end{cases}.
$$

Let $Y_{1}=X_{1} sgn\left(X_{2}\right)$ and $Y_{2}=X_{2} sgn\left(X_{1}\right)$. If the correlation coefficient between $Y_{1}$ and $Y_{2}$ is $\alpha$,
then $\pi \alpha$ is equal to ____

Answer: 2

Some Useful Links:

• IIT JAM MS (Set B) 2021 Question Paper - Problems and Solutions
• IIT JAM MS (Set A) 2021 Question Paper - Problems and Solutions
• How to Prepare for IIT JAM Statistics?
• Know about the learning Paths
• Our Statistics Program

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.

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

Problem 1

The value of the limit

$$
\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\left(\begin{array}{c}
2 n \\
k
\end{array}\right) \frac{1}{4^{n}}
$$

is equal to

Options-

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $1$
4. $0$

Answer: $\frac{1}{2}$

Problem 2

If the series $\sum_{n=1}^{\infty} a_{n}$ converges absolutely, then which of the following series diverges?

Options-

1. $\sum_{n=1}^{\infty}\left|a_{2 n}\right|$
2. $\sum_{n=1}^{\infty}\left(a_{n}\right)^{3}$
3. $\sum_{n=2}^{\infty}\left(\frac{1}{(\ln n)^{2}}+a_{n}\right)$
4. $\sum_{n=1}^{\infty} \frac{a_{n}+a_{n+1}}{2}$

Answer: $\sum_{n=2}^{\infty}\left(\frac{1}{(\ln n)^{2}}+a_{n}\right)$

Problem 3

Let $X$ be a $U(0,1)$ random variable and let $Y=X^{2}$. If $\rho$ is the correlation coefficient between the random variables $X$ and $Y$, then $48 \rho^{2}$ is equal to

Options-

1.$48$;
2.$30$;
3.$45$;
4.$35$.

Answer: $45$
Solution:

Problem 4

Let $\{X_{n}\}_{n \geq 1}$ be a sequence of independent and identically distributed random variables with probability density function

$$f(x)=\begin{cases} 1, &amp; \text { if } 0&lt;x&lt;1 \\ 0, &amp; \text { otherwise } \end{cases}$$

Then. the value of the limit

$$\lim _{n \rightarrow \infty} P\left(-\frac{1}{n} \sum{i=1}^{n} \ln X_{i} \leq 1+\frac{1}{\sqrt{n}}\right)$$ is equal to

Options -

1. $0$;
2. $\Phi(2)$;
3. $\Phi(1)$;
4. $\frac{1}{2}$.

Answer: $\Phi(1)$

Problem 5

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

$$
f(x)=x^{7}+5 x^{3}+11 x+15, x \in \mathbb{R}
$$

Then, which of the following statements is TRUE?

Options -

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

Answer: $f$ is both one-one and onto

Problem 6

There are three urns, labeled. Urn $1$ , Urn $2$ and Urn $3$ . Urn $1$ contains $2$ white balls and $2$ black balls, Urn $2$ contains $1$ white ball and $3$ black balls and Urn $3$ contains $3$ white balls and $1$ black ball. Consider two coins with probability of obtaining head in their single trials as $0.2$ and $0.3 .$ The two coins are tossed independently once, and an urn is selected according to the following scheme:
Urn $1$ is selected if $2$ heads are obtained: Urn $3$ is selected if $2$ tails are obtained; otherwise Urn $2$ is
selected. A ball is then drawn at random from the selected urn. Then
$P($ Urn 1 is selected $\mid$ the ball drawn is white $)$ is equal to

Options -

1. $\frac{12}{109}$
2. $\frac{1}{18}$
3. $\frac{6}{109}$
4. $\frac{1}{9}$

Answer: $\frac{6}{109}$

Problem 7

Let $X$ be a random variable with probability density function

$$
f(x)=\frac{1}{2} e^{-|x|}, \quad-\infty<x<\infty
$$

Then, which of the following statements is FALSE?

Options -

1. $E\left(|X| \sin \left(\frac{X}{|X|}\right)\right)=0$
2. $E(X|X|)=0$
3. $E\left(|X| \sin ^{2}\left(\frac{X}{|X|}\right)\right)=0$
4. $E\left(X|X|^{2}\right)=0$

Answer: $E\left(|X| \sin ^{2}\left(\frac{X}{|X|}\right)\right)=0$

Problem 8

The value of the limit

.

$$
\lim _{n \rightarrow \infty}\left(\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \cdots\left(1+\frac{n}{n}\right)\right)^{\frac{1}{n}}
$$

is equal to

Options-

1. $\frac{3}{e}$
2. $\frac{4}{e}$
3. $e$
4. $\frac{1}{e}$

Answer: $\frac{4}{e}$
Solution:

Problem 9

Let $M$ be a $3 \times 3$ real matrix. Let $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}0 \\ -1 \\ \alpha\end{array}\right)$ be the eigenvectors of $M$ corresponding to three distinct eigenvalues of $M$. where $\alpha$ is a real number. Then. which of the following is NOT a possible value of $\alpha$ ?

Options-

1.$1$
2 .$-2$
3.$2$
4.$0$

Answer: $-2$

Problem 10

The value of the limit

$$
\lim _{x \rightarrow 0} \frac{e^{-3 x}-e^{x}+4 x}{5(1-\cos x)}
$$ is equal to

Options -

1. $\frac{2}{5}$
2. 0
3. $\frac{8}{5}$
4. 1

Answer: $\frac{8}{5}$

Problem 11

Consider a sequence of independent Bernoulli trials with probability of success in each trial as $\frac{1}{3}$. The probability that three successes occur before four failures is equal to

Options -

1.$\frac{179}{841}$
2.$\frac{179}{243}$
3.$\frac{233}{729}$
4.$\frac{179}{1215}$

Answer: $\frac{179}{1215}$

Problem 12

Let,

$$
S=\sum_{k=1}^{\infty}(-1)^{k-1} \frac{1}{k}\left(\frac{1}{4}\right)^{k} \text { and } T=\sum_{k=1}^{\infty} \frac{1}{k}\left(\frac{1}{5}\right)^{k}
$$

Then, which of the following statements is TRUE?

Options -

1.$5 S-4 T=0$
2.$S-T=0$

3. $16 S-25 T=0$
4. $4 S-5 T=0$

Answer: $S-T=0$

IIT JAM 2021 - Problem 13

Let $a_{1}=5$ and define recursively

$$
a_{n+1}=3^{\frac{1}{4}}\left(a_{n}\right)^{\frac{3}{4}}, \quad n \geq 1
$$

Then, which of the following statements is TRUE?

Options-

1. $\{a_{n}\}$ is monotone decreasing, and $\lim _{n \rightarrow \infty} a{n}=3$
2. $\{a_{n}\}$ is decreasing, and $\lim_ {n \rightarrow \infty} a{n}=0$
3. $\{a_{n}\}$ is non-monotone, and $\lim_ {n \rightarrow \infty} a{n}=3$
4. $\{a_{n}\}$ is monotone increasing, and $\lim _{n \rightarrow \infty} a{n}=3$

Answer:$\{a_{n}\}$ is monotone decreasing, and $\lim _{n \rightarrow \infty} a{n}=3$

Problem 14

Let $E_{1}, E_{2}$ and $E_{3}$ be three events such that $P\left(E_{1}\right)=\frac{4}{5}, P\left(E_{2}\right)=\frac{1}{2}$ and $P\left(E_{3}\right)=\frac{9}{10}$
Then. which of the following statements is FALSE?

1. $P\left(E_{1} \cup E_{2} \cup E_{3}\right) \geq \frac{9}{10}$
2. $P\left(E_{1} \cup E_{2}\right) \geq \frac{4}{5}$
3. $P\left(E_{2} \cap E_{3}\right) \leq \frac{1}{2}$
4. $P\left(E_{1} \cap E_{2} \cap E_{3}\right) \leq \frac{1}{6}$

Answer: $P\left(E_{1} \cap E_{2} \cap E_{3}\right) \leq \frac{1}{6}$

Problem 15

Let $E_{1}, E_{2}, E_{3}$ and $E_{4}$ be four events such that
$$
P\left(E_{i} \mid E_{4}\right)=\frac{2}{3}, i=1,2,3 ; P\left(E_{i} \cap E_{j}^{c} \mid E_{4}\right)=\frac{1}{6}, i, j=1,2,3 ; i \neq j \text { and } P\left(E_{1} \cap E_{2} \cap E_{3}^{c} \mid E_{4}\right)=\frac{1}{6}
$$
Then. $P\left(E_{1} \cup E_{2} \cup E_{3} \mid E_{4}\right)$ is equal to

1. $\frac{1}{2}$
2. $\frac{5}{6}$
3. $\frac{2}{3}$
4. $\frac{7}{12}$

Answer: $\frac{5}{6}$

Problem 16

Let $X$ be a random variable having the probability density function

$$
f(x)=\begin{cases}
e^{-x}, & x>0 \\
0, & x \leq 0
\end{cases}.
$$

Define $Y=[X]$, where $[X]$ denotes the largest integer not exceeding $X$. Then, $E\left(Y^{2}\right)$ is equal to

Options -

1.$\frac{e+1}{(e-1)^{2}}$

2. $\frac{(e+1)^{2}}{(e-1)^{2}}$
3. $\frac{e(e+1)^{2}}{e-1}$
4. $\frac{e(e+1)}{e-1}$

Answer: $\frac{e+1}{(e-1)^{2}}$

Problem 17

Let $X$ be a continuous random variable with distribution function

$$
F(x)=\begin{cases}
0, & \text { if } x<0 \\
a x^{2}, & \text { if } 0 \leq x<2 \\
1, & \text { if } x \geq 2
\end{cases}.
$$

for some real constant $a$. Then, $E(X)$ is equal to

Options -

1.1
2 .$ \frac{4}{3}$

3. $\frac{1}{4}$
4. 0

Answer: $ \frac{4}{3}$

Problem 18

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 2)$ be a random sample from $U(\theta-5, \theta+5),$ where $\theta \in(0, \infty)$ is unknown. Let $T=\max \{X_{1}, X_{2}, \ldots, X_{n}\}$ and $U=\min \{X_{1}, X_{2}, \ldots, X_{n}\} .$ Then, which of the following statements is TRUE?

Options -

1. $U+8$ is an MLE of $\theta$
2. $\frac{T+U}{2}$ is the unique $\mathrm{MLE}$ of $\theta$
3. MLE of $\frac{1}{\theta}$ does NOT exist
4. $\frac{2}{T+U}$ is an $\mathrm{MLE}$ of $\frac{1}{\theta}$

Answer: $\frac{2}{T+U}$ is an $\mathrm{MLE}$ of $\frac{1}{\theta}$

Problem 19

Consider the problem of testing $H_{0}: X \sim f_{0}$ against $H_{1}: X \sim f_{1}$ based on a sample of size 1 , where

$f_{0}(x)=\begin{cases}1, 0 \leq x \leq 1 \\ 0, \text { otherwise }\end{cases}.$ and $f_{1}(x)=\begin{cases}2-2 x , 0 \leq x \leq 1 \\ 0, \text { otherwise }\end{cases}$.

Then, the probability of Type II error of the most powerful test of size $\alpha=0.1$ is equal to

Options -

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

Answer: 0.81
Solution:

Problem 20

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

Options -

1. $P(X<6)>P(Y<6)$
2. $P(X>0.7)>P(Y>0.7)$

3. $P(X>3)3)$
4. $P(X>0.7)0.7)$

Answer: $P(X>0.7)>P(Y>0.7)$

Problem 21

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by

$f(x, y)=\begin{cases}\frac{y^{3}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \ 0, & (x, y)=(0,0)\end{cases}$.

Let $f_{x}(x, y)$ and $f_{y}(x, y)$ denote the first order partial derivatives of $f(x, y)$ with respect to $x$ and $y$,
respectively, at the point $(x, y)$. Then, which of the following statements is FALSE?

Options -

1. $f$ is NOT differentiable at (0,0)
2. $f_{y}(0,0)$ exists and $f_{y}(x, y)$ is continuous at (0,0)
3. $f_{y}(x, y)$ exists and is bounded at every $(x, y) \in \mathbb{R}^{2}$
4. $f_{x}(x, y)$ exists and is bounded at every $(x, y) \in \mathbb{R}^{2}$

Answer: $f_{y}(0,0)$ exists and $f_{y}(x, y)$ is continuous at (0,0)

Problem 22

Let $(X, Y)$ be a random vector with joint moment generating function

$$
M\left(t_{1}, t_{2}\right)=\frac{1}{\left(1-\left(t_{1}+t_{2}\right)\right)\left(1-t_{2}\right)}, \quad-\infty<t_{1}<\infty,-\infty<t_{2}<\min \{1,1-t_{1}\}
$$

Let $Z=X+Y$. Then. $Var(Z)$ is equal to

Options -

1.3

2.4

3.5

4.6

Answer: 5

Problem 23

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from an exponential distribution with probability density function

$$
f(x ; \theta)=\begin{cases}
\theta e^{-\theta x}, x>0 \\
0, \text { otherwise }
\end{cases}
$$

where $\theta \in(0, \infty)$ is unknown. Let $\alpha \in(0,1)$ be fixed and let $\beta$ be the power of the most powerful test of size $\alpha$ for testing $H_{0}: \theta=1$ against $H_{1}: \theta=2$.
Consider the critical region

$R=\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n} ; \sum_{l=1}^{n} x_{i}>\frac{1}{2} \chi_{2 n}^{2}(1-\alpha)\}$

where for any $\gamma \in(0,1), \chi_{2 n}^{2}(\gamma)$ is a fixed point such that $P\left(x_{2 n}^{2}>x_{2 n}^{2}(\gamma)\right)=\gamma .$ Then, the
critical region $R$ corresponds to the

Options-

1.most powerful test of size $\beta$ for testing $H_{0}^{}: \theta=2$ against $H_{1}^{}: \theta=1$
2.most powerful test of size $\alpha$ for testing $H_{0}: \theta=1$ against $H_{1}: \theta=2$
3.most powerful test of size $1-\beta$ for testing $H_{0}^{}: \theta=2$ against $H_{1}^{}: \theta=1$
4.most powerful test of size $1-\alpha$ for testing $H_{0}^{}: \theta=2$ against $H_{1}^{}: \theta=1$

Answer: most powerful test of size $\alpha$ for testing $H_{0}^{}: \theta=2$ against $H_{1}^{}: \theta=1$ [No Option]

Problem 24

Consider three coins having probabilities of obtaining head in a single trial as $\frac{1}{4}, \frac{1}{2}$ and $\frac{3}{4}$, 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 -

1. $\frac{64}{384}$
2. $\frac{125}{384}$
3. $\frac{255}{384}$
4. $\frac{85}{384}$

Answer: $\frac{85}{384}$

Problem 25

For $a \in \mathbb{R}$, consider the system of linear equations

$\begin{array}{ll}a x+a y & =a+2 \\ x+a y+(a-1) z & =a-4 \\ a x+a y+(a-2) z & =-8\end{array}$

in the unknowns $x, y$ and $z$. Then. which of the following statements is $\mathbf{T R U E}$ ?

Options -

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

Answer: The given system has a unique solution for $a=-2$

Problem 26

Let $X$ and $Y$ be independent $N(0,1)$ random variables and $Z=\frac{|X|}{|Y|} .$ Then, which of the
following expectations is finite?

Options -

1. $E(Z)$
2. $E\left(\frac{1}{Z \sqrt{Z}}\right)$
3. $E(Z \sqrt{Z})$
4. $E\left(\frac{1}{\sqrt{Z}}\right)$

Answer: $E\left(\frac{1}{\sqrt{Z}}\right)$

Problem 27

Let $\{X_{n}\}_{n>1}$ be a sequence of independent and identically distributed $N(0,1)$ random variables.
Then,

$$
\lim _{n \rightarrow \infty} P\left(\frac{\sum{i=1}^{n} X_{i}^{4}-3 n}{\sqrt{32 n}} \leq \sqrt{6}\right)
$$ is equal to

Options -

1. 0
2. $\Phi(\sqrt{2})$
3. $\frac{1}{2}$
4. $\Phi(1)$

Answer: $\Phi(\sqrt{2})$

Problem 28

Let $X$ be a continuous random variable having the moment generating function

$$
M(t)=\frac{e^{t}-1}{t}, \quad t \neq 0
$$

Let $\alpha=P\left(48 X^{2}-40 X+3>0\right)$ and $\beta=P\left((\ln X)^{2}+2 \ln X-3>0\right)$.
Then, the value of $\alpha-2 \ln \beta$ is equal to

Options-

1. $\frac{10}{3}$
2. $\frac{13}{3}$
3. $\frac{19}{3}$
4. $\frac{17}{3}$

Answer: $\frac{19}{3}$

Problem 29

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 3)$ be a random sample from Poisson $(\theta),$ where $\theta \in(0, \infty)$ is unknown and
let

$$
T=\sum_{i=1}^{n} X_{i}
$$

Then, the uniformly minimum variance unbiased estimator of $e^{-2 \theta} \theta^{3}$

Options -

1. is $\quad \frac{T}{n}\left(\frac{T}{n}-1\right)\left(\frac{T}{n}-2\right)\left(1-\frac{2}{n}\right)^{T-3}$
2. is $\frac{T(T-1)(T-2)(n-2)^{T-3}}{n^{T}}$
3. does NOT exist
4. is $e^{-\frac{2 T}{n}\left(\frac{T}{n}\right)^{3}}$

Answer: $\frac{T(T-1)(T-2)(n-2)^{T-3}}{n^{T}}$

Problem 30

Let $\{a_{n}\}_{n \geq 1}$ be a sequence of real numbers such that $a_{n} \geq 1$, for all $n \geq 1$. Then, which of the following conditions imply the divergence of $\{a_{n}\}_{n \geq 1} ?$

Options -

1$\sum_{n=1}^{\infty} b_{n}$ converges, where $b_{1}=a_{1}$ and $b_{n}=a_{n+1}-a_{n},$ for all $n>1$

2. $\{\sqrt{a_{n}}\}_{n \geq 1}$ converges
3. $\lim _{n \rightarrow \infty} \frac{a{2 n+1}}{a_{2 n}}=\frac{1}{2}$
4. $\{a_{n}\}_{n} \geq 1$ is non-increasing

Answer: $\lim _{n \rightarrow \infty} \frac{a{2 n+1}}{a_{2 n}}=\frac{1}{2}$

Some Useful Links:

• IIT JAM MS (Set B) 2021 Question Paper - Problems and Solutions
• IIT JAM MS (Set C) 2021 Question Paper - Problems and Solutions
• How to Prepare for IIT JAM Statistics?
• Know about the learning Paths
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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.

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

Problem 1

A sample of size $n$ is drawn randomly (without replacement) from an urn couraining $5 n^{2}$ balls, of which $2 n^{2}$ are red balls and $3 n^{2}$ are black balls. Let $X_{n}$ denote the number of red balls in the selected sample. If $\ell=\lim _{n \rightarrow \infty} \frac{E\left(X{n}\right)}{n}$ and $m=\lim _{n \rightarrow \infty} \frac{Var (X{n})}{n},$ then which of the following statements is/are TR UE?

Options -

1. $\frac{\ell}{m}=\frac{5}{3}$
2. $\ell m=\frac{14}{125}$
3. $\ell-m=\frac{3}{25}$
4. $\ell+m=\frac{16}{25}$

Answer: $\frac{\ell}{m}=\frac{5}{3}$; $\ell+m=\frac{16}{25}$

Problem 2

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 2)$ be a random sample from a distribution with probability density function

$$
f(x ; \theta)=\begin{cases}
\frac{3 x^{2}}{\theta} e^{-x^{3} / \theta}, x>0 \\
0, \text { othervise }
\end{cases}.
$$

where $\theta \in(0, \infty)$ is unknown.
If $T=\sum_{i =1}^{n} X_{i}^{3}$, then which of the following statements is/are TRUE?

Options -

1.$\frac{n-1}{T}$ is the unique uniformly minimum variance unbiased estimator of $\frac{1}{\theta}$
2.$\frac{n}{T}$ is the unique uniformly minimum variance unbiased estimator of $\frac{1}{\theta}$
3.$(n-1) \sum_{i=1}^{n} \frac{1}{x_{i}^{3}}$ is the unique uniformly minimum variance unbiased estimator of $\frac{1}{\theta}$

4. $\frac{n}{T}$ is the MLE of $\frac{1}{\theta}$

Answer:
$\frac{n-1}{T}$ is the unique uniformly minimum variance unbiased estimator of $\frac{1}{\theta}$
$\frac{n}{T}$ is the MLE of $\frac{1}{\theta}$

Problem 3

Consider the linear system $A \underline{x}=\underline{b}$, where $A$ is an $m \times n$ matrix, $\underline{x}$ is an $n \times 1$ vector of unknowns
and $b$ is an $m \times 1$ vector. Further, suppose there exists an $m \times 1$ vector $c$ such that the linear system $A \underline{x}=c$ has No solution. Then, which of the following statements is/are necessarily TRUE?

Options -

1.If $m \leq n$ and $d$ is the first column of $A$, then the linear system $A \underline{x}=\underline{d}$ has a unique solution
2.If $m>n,$ then the linear system $A x=0$ has a solution other than $x=0$

3. If $m \geq n,$ then $Rank(A)<n$
4. $Rank(A)<m$

.
Answer:
$Rank(A)<m$

Problem 4

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 2)$ be independent and identically distributed random variables with probability density function

$$
f(x)=\begin{cases}
\frac{1}{x^{2}}, x \geq 1 \\
0, \text { otherwise }
\end{cases}.
$$

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

Options -

1. $\frac{1}{X_{2}}$
2. $\sqrt{X_{1}}$
3. $X_{1}$
4. $\min \{X_{1}, \ldots, X_{n}\}$

Answer: $\frac{1}{X_{2}}$, $\sqrt{X_{1}}$, $\min \{X_{1}, \ldots, X_{n}\}$

Problem 5

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from $N(\theta, 1),$ where $\theta \in(-\infty, \infty)$ is unknown. Consider the problem of testing $H_{0}: \theta \leq 0$ against $H_{1}: \theta>0 .$ Let $\beta(\theta)$ denote the power function of the likelihood ratio test of size $\alpha(0<\alpha<1)$ for testing $H_{0}$ against $H_{1}$. Then. which of the following statements is/are TRUE?

Options -

1.The critical region of the likelihood test of size $\alpha$ is
$$
\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}: \sqrt{n} \frac{\sum_{i=1}^{n} x_{i}}{n}<\tau_{\alpha}\} $$ where $\tau_{\alpha}$ is a fixed point such that $P\left(Z>\tau_{\alpha}\right)=\alpha, Z \sim N(0,1)$

2. $\beta(\theta)>\beta(0),$ for all $\theta>0$
3. The critical region of the likelihood test of size $\alpha$ is
   $$\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}: \sqrt{n} \frac{\sum_{i=1}^{n} x_{i}}{n}>\tau_{\alpha / 2}\}$$
   where $\tau_{\alpha / 2}$ is a fixed point such that $P\left(Z>\tau_{\alpha / 2}\right)=\frac{\alpha}{2}, Z \sim N(0,1)$
4. $\beta(\theta)<\beta(0),$ for all $\theta>0$

Answer: $\beta(\theta)>\beta(0),$ for all $\theta>0$

Problem 6

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 2)$ be a random sample from a distribution with probability density function

$$
f(x ; \theta)=\begin{cases}
\frac{1}{2 \theta}, -\theta \leq x \leq \theta \\
0, |x|>\theta
\end{cases}.
$$

where $\theta \in(0, \infty)$ is unknown. If $R=\min \{X_{1}, X_{2}, \ldots, X_{n}\}$ and $S=\max \{X_{1}, X_{2}, \ldots, X_{n}\},$ then which
of the following statements is/are TRUE?

Options -

1.$\max \{\left|X_{1}\right|,\left|X_{2}\right|, \ldots,\left|X_{n}\right|\}$ is a complete and sufficient statistic for $\theta$

2. $S$ is an $\mathrm{MLE}$ of $\theta$
3. $(R, S)$ is jointly sufficient for $\theta$
4. Distribution of $\frac{R}{S}$ does NOT depend on $\theta$

Answer:
$\max \{\left|X_{1}\right|,\left|X_{2}\right|, \ldots,\left|X_{n}\right|\}$ is a complete and sufficient statistic for $\theta$
$(R, S)$ is jointly sufficient for $\theta$
Distribution of $\frac{R}{S}$ does NOT depend on $\theta$

Problem 7

Let $X_{1}, X_{2}, \ldots, X_{n}(n \geq 2)$ be a random sample from a distribution with probability density function

$$
f(x ; \theta)=\begin{cases}
\theta x^{\theta-1}, 0 \leq x \leq 1 \\
0, \text { otherwise }
\end{cases}.
$$

where $\theta \in(0, \infty)$ is unknown. Then, which of the following statements is/are TRUE?

Options -

1 .There does NOT exist any unbiased estimator of $\frac{1}{\theta}$ which attains the Cramer-Rao lower bound
2.Cramer-Rao lower bound, based on $X_{1}, X_{2}, \ldots, X_{n},$ for the estimand $\theta^{3}$ is $\frac{\theta^{2}}{n}$
3 .Cramer-Rao lower bound. based on $X_{1}, X_{2}, \ldots, X_{n},$ for the estimand $\theta^{3}$ is $9 \frac{\theta^{6}}{n}$
4 .There exists an unbiased estimator of $\frac{1}{\theta}$ which attains the Cramer-Rao lower bound

Answer:
Cramer-Rao lower bound. based on $X_{1}, X_{2}, \ldots, X_{n},$ for the estimand $\theta^{3}$ is $9 \frac{\theta^{6}}{n}$
There exists an unbiased estimator of $\frac{1}{\theta}$ which attains the Cramer-Rao lower bound

Problem 8

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?

Options-

1. $f^{\prime \prime}$ is continuous
2. $f^{\prime \prime}$ is bounded on (0,1)
3. If $f^{\prime}(0)=f^{\prime}(1),$ then $f^{\prime \prime}(x)=0$ has a solution in (0,1)
4. $f^{\prime}$ is bounded on [8,10]

Answer:
If $f^{\prime}(0)=f^{\prime}(1),$ then $f^{\prime \prime}(x)=0$ has a solution in (0,1)
$f^{\prime}$ is bounded on [8,10]

Problem 9

Let $A$ be a $3 \times 3$ real matrix such that $A \neq I_{3}$ and the sum of the entries in each row of $A$ is $1$. Then which of the following statements is/are necessarily TRUE?

Options -

1. The characteristic polynomial, $p(\lambda),$ of $A+2 A^{2}+A^{3}$ has $(\lambda-4)$ as a factor

2. $A-I_{3}$ is an invertible matrix
3. $A$ cannot be an orthogonal matrix
4. The set $\{\underline{x} \in \mathbb{R}^{3}:\left(A-I_{3}\right) \underline{x}=\underline{0}\}$ has at least two elements $(\underline{x}$ is a column vector)

Answer:
The characteristic polynomial, $p(\lambda),$ of $A+2 A^{2}+A^{3}$ has $(\lambda-4)$ as a factor

Problem 10

Consider the function

$$
f(x, y)=3 x^{2}+4 x y+y^{2}, \quad(x, y) \in \mathbb{R}^{2}
$$

If $S=\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\}$, then which of the following statements is/are TRUE?

Options -

1. The maximum value of $f$ on $S$ is $2+\sqrt{5}$
2. The maximum value of $f$ on $S$ is $3+\sqrt{5}$
3. The minimum value of $f$ on $S$ is $3-\sqrt{5}$
4. The minimum value of $f$ on $S$ is $2-\sqrt{5}$

Answer:
The maximum value of $f$ on $S$ is $2+\sqrt{5}$
The minimum value of $f$ on $S$ is $2-\sqrt{5}$

Some Useful Links:

• IIT JAM MS (Set A) 2021 Question Paper - Problems and Solutions
• IIT JAM MS (Set C) 2021 Question Paper - Problems and Solutions
• How to Prepare for IIT JAM Statistics?
• Know about the learning Paths
• Our Statistics Program