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IIT JAM MS 2021 Question Paper | Set C | Problems & Solutions

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:


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

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