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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

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

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)={1,0, if 0<x<1 otherwise

Then. the value of the limit

limn→∞P(1ni=1nlnXi≤1+1n)
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.$S-T=0$

1. $16 S-25 T=0$
2. $4 S-5 T=0$

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

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

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 .$\frac{4}{3}$

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

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. $P(X>0.7)>P(Y>0.7)$
1. $P(X>3)3)$
2. $P(X>0.7)0.7)$

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

## 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 $n>1$

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

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

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

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

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)={1,0, if 0<x<1 otherwise

Then. the value of the limit

limn→∞P(1ni=1nlnXi≤1+1n)
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.$S-T=0$

1. $16 S-25 T=0$
2. $4 S-5 T=0$

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

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

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 .$\frac{4}{3}$

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

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. $P(X>0.7)>P(Y>0.7)$
1. $P(X>3)3)$
2. $P(X>0.7)0.7)$

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

## 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 $n>1$

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

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

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