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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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 -
Answer: $\frac{\ell}{m}=\frac{5}{3}$; $\ell+m=\frac{16}{25}$
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 $\frac{1}{\theta}$
Answer:
$\frac{n-1}{T}$ is the unique uniformly minimum variance unbiased estimator of $\frac{1}{\theta}$
$\frac{n}{T}$ is the MLE of $\frac{1}{\theta}$
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 $x=0$
.
Answer:
$Rank(A)<m$
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 -
Answer: $\frac{1}{X_{2}}$, $\sqrt{X_{1}}$, $\min \{X_{1}, \ldots, X_{n}\}$
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 $P\left(Z>\tau_{\alpha}\right)=\alpha, Z \sim N(0,1)$
Answer: $\beta(\theta)>\beta(0),$ for all $\theta>0$
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 $\theta$
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 $\theta$
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
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?
Options-
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]
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 -
Answer:
The characteristic polynomial, $p(\lambda),$ of $A+2 A^{2}+A^{3}$ has $(\lambda-4)$ as a factor
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 -
Answer:
The maximum value of $f$ on $S$ is $2+\sqrt{5}$
The minimum value of $f$ on $S$ is $2-\sqrt{5}$
