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.
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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: $13$
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
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: $7$
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
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
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
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
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
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: $6$
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: $8$
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
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
The number of real roots of the polynomial
$$
f(x)=x^{11}-13 x+5
$$
is ____
Answer:$3$
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
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
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
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
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
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
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
