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
$$
\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\left(\begin{array}{c}
2 n \\
k
\end{array}\right) \frac{1}{4^{n}}
$$
is equal to
Options-
Answer: $\frac{1}{2}$
If the series $\sum_{n=1}^{\infty} a_{n}$ converges absolutely, then which of the following series diverges?
Options-
Answer: $\sum_{n=2}^{\infty}\left(\frac{1}{(\ln n)^{2}}+a_{n}\right)$
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:
Let $\{X_{n}\}_{n \geq 1}$ be a sequence of independent and identically distributed random variables with probability density function
Then. the value of the limit
Options -
Answer: $\Phi(1)$
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 -
Answer: $f$ is both one-one and onto
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 -
Answer: $\frac{6}{109}$
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 -
Answer: $E\left(|X| \sin ^{2}\left(\frac{X}{|X|}\right)\right)=0$
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-
Answer: $\frac{4}{e}$
Solution:
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: $-2$
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 -
Answer: $\frac{8}{5}$
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: $\frac{179}{1215}$
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.$S-T=0$
Answer: $S-T=0$
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-
Answer:$\{a_{n}\}$ is monotone decreasing, and $\lim _{n \rightarrow \infty} a{n}=3$
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?
Answer: $P\left(E_{1} \cap E_{2} \cap E_{3}\right) \leq \frac{1}{6}$
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
Answer: $\frac{5}{6}$
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.$\frac{e+1}{(e-1)^{2}}$
Answer: $\frac{e+1}{(e-1)^{2}}$
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}$
Answer: $ \frac{4}{3}$
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 -
Answer: $\frac{2}{T+U}$ is an $\mathrm{MLE}$ of $\frac{1}{\theta}$
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 -
Answer: 0.81
Solution:
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 -
Answer: $P(X>0.7)>P(Y>0.7)$
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 -
Answer: $f_{y}(0,0)$ exists and $f_{y}(x, y)$ is continuous at (0,0)
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
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]
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 -
Answer: $\frac{85}{384}$
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 -
Answer: The given system has a unique solution for $a=-2$
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 -
Answer: $E\left(\frac{1}{\sqrt{Z}}\right)$
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 -
Answer: $\Phi(\sqrt{2})$
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-
Answer: $\frac{19}{3}$
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 -
Answer: $\frac{T(T-1)(T-2)(n-2)^{T-3}}{n^{T}}$
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 $n>1$
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.
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-
Answer: $\frac{1}{2}$
If the series $\sum_{n=1}^{\infty} a_{n}$ converges absolutely, then which of the following series diverges?
Options-
Answer: $\sum_{n=2}^{\infty}\left(\frac{1}{(\ln n)^{2}}+a_{n}\right)$
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:
Let $\{X_{n}\}_{n \geq 1}$ be a sequence of independent and identically distributed random variables with probability density function
Then. the value of the limit
Options -
Answer: $\Phi(1)$
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 -
Answer: $f$ is both one-one and onto
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 -
Answer: $\frac{6}{109}$
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 -
Answer: $E\left(|X| \sin ^{2}\left(\frac{X}{|X|}\right)\right)=0$
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-
Answer: $\frac{4}{e}$
Solution:
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: $-2$
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 -
Answer: $\frac{8}{5}$
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: $\frac{179}{1215}$
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.$S-T=0$
Answer: $S-T=0$
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-
Answer:$\{a_{n}\}$ is monotone decreasing, and $\lim _{n \rightarrow \infty} a{n}=3$
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?
Answer: $P\left(E_{1} \cap E_{2} \cap E_{3}\right) \leq \frac{1}{6}$
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
Answer: $\frac{5}{6}$
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.$\frac{e+1}{(e-1)^{2}}$
Answer: $\frac{e+1}{(e-1)^{2}}$
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}$
Answer: $ \frac{4}{3}$
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 -
Answer: $\frac{2}{T+U}$ is an $\mathrm{MLE}$ of $\frac{1}{\theta}$
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 -
Answer: 0.81
Solution:
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 -
Answer: $P(X>0.7)>P(Y>0.7)$
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 -
Answer: $f_{y}(0,0)$ exists and $f_{y}(x, y)$ is continuous at (0,0)
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
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]
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 -
Answer: $\frac{85}{384}$
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 -
Answer: The given system has a unique solution for $a=-2$
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 -
Answer: $E\left(\frac{1}{\sqrt{Z}}\right)$
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 -
Answer: $\Phi(\sqrt{2})$
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-
Answer: $\frac{19}{3}$
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 -
Answer: $\frac{T(T-1)(T-2)(n-2)^{T-3}}{n^{T}}$
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 $n>1$
Answer: $\lim _{n \rightarrow \infty} \frac{a{2 n+1}}{a_{2 n}}=\frac{1}{2}$
Please claim problem 11 to official answer key