Figure from the original question paper Figure from the original question paper Figure from the original question paper Figure from the original question paper Figure from the original question paper

(A) Useful Data has been mentioned above.

(B) This data attachment is just for a reference & not for an evaluation.

The joint probability density function of two continuous random variables and is given as, Find the marginal density of . The joint probability density function of two continuous random variables and is given as,

(B) Figure from the original question paper

(A) Figure from the original question paper

(C) Figure from the original question paper

(D) Figure from the original question paper

The probability density function of a continuous random variable is given by Based on the above data, answer the given subquestions. The probability density function of a continuous random variable is given by Find the expected value of .

(D) Figure from the original question paper

(A) Figure from the original question paper

(B) Figure from the original question paper

(C) Figure from the original question paper

Stats 2 End Term PYQ May 2025: 14 Sept 2025 (Set 1) · IITM BS

Question 1

+1 markOne correct option

A
Useful Data has been mentioned above.
B
This data attachment is just for a reference & not for an evaluation.

Show answer

Correct answer

A
Useful Data has been mentioned above.

Question 1 on its own page

Question 2

+3 marksWritten answer

A supermarket manager claims that the average billing time for customers is 12 minutes with a known standard deviation of 3.5 minutes. To verify this claim, a random sample of 49 customers was studied, showing an average billing time of 12.9 minutes. Let the null and alternative hypothesis be: ; . Find the P-value. Enter the answer correct to two decimal places.

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Question 2 on its own page

Question 3

+3 marksOne correct option

Consider the samples {1, 3, 4, 6, 2} taken from a Poisson Distribution with an unknown parameter . Using a Gamma(5, 8) prior, find the posterior distribution of .

A
Gamma
B
Gamma
C
Gamma
D
Gamma

Show answer

Correct answer

D
Gamma

Question 3 on its own page

Question 4

+3 marksOne correct option

The joint probability density function of two continuous random variables and is given as,

Find the marginal density of .

A
B
C
D

Show answer

Correct answer

B

Question 4 on its own page

Question 5

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Let be a discrete random variable with the probability mass function

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 5 on its own page

Question 6

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 6 on its own page

Question 7

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 7 on its own page

Question 8

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 8 on its own page

Question 9

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 9 on its own page

Question 10

+0.5 marksWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 10 on its own page

Question 11

+3 marksWritten answer

A nutrition company claims that the average protein content of its energy bars is 20 grams, with a known standard deviation of 4 grams. A sports academy suspects that the actual mean protein content may be less than 20 grams. To test the claim, the academy collects a random sample of 64 bars. The academy decides to reject the company’s claim if the mean protein content of the sample and accepts it otherwise. Assuming that the company’s claim is true, what is the probability that the academy rejects the claim?

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Question 11 on its own page

Question 12

+2 marksWritten answer

A laboratory is testing a new temperature sensor to check whether it provides more consistent readings than the old model. The lab records the temperature (in oC) from 5 repeated measurements using the new sensor: 10, 8, 2, 5, 15 The old temperature sensor has a known standard deviation of . Based on the above data, answer the given subquestions.

Compute the sample variance of the 5 temperature readings from the new sensor. Enter the answer correct to one decimal place.

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A written answer, not marked automatically.

Question 12 on its own page

Question 13

+3 marksOne correct option

A laboratory is testing a new temperature sensor to check whether it provides more consistent readings than the old model. The lab records the temperature (in oC) from 5 repeated measurements using the new sensor: 10, 8, 2, 5, 15 The old temperature sensor has a known standard deviation of . Based on the above data, answer the given subquestions.

At significance level, determine whether the new sensor has less variability than the old sensor.

A
The new sensor does not show less variability than the old sensor.
B
The new sensor has less variability than the old sensor.

Show answer

Correct answer

A
The new sensor does not show less variability than the old sensor.

Question 13 on its own page

Question 14

+3 marksOne correct option

A factory manufactures light bulbs, and each bulb has a probability of being defective. To estimate this probability using Bayesian methods, assume the prior distribution of to be Beta(3, 3). A quality inspector tests 12 independent bulbs. If 5 bulbs are found to be defective, answer the given subquestions.

Determine the posterior distribution of .

A
Beta
B
Beta
C
Beta
D
Beta

Show answer

Correct answer

D
Beta

Question 14 on its own page

Question 15

+2 marksWritten answer

A factory manufactures light bulbs, and each bulb has a probability of being defective. To estimate this probability using Bayesian methods, assume the prior distribution of to be Beta(3, 3). A quality inspector tests 12 independent bulbs. If 5 bulbs are found to be defective, answer the given subquestions.

Using the posterior distribution, calculate the posterior mean of . Enter the answer correct to two decimal places.

Show answer

A written answer, not marked automatically.

Question 15 on its own page

Question 16

+3 marksWritten answer

The probability density function of a continuous random variable is given by

Based on the above data, answer the given subquestions.

Let be the CDF of . Calculate the value of . Enter the answer correct to two decimal places.

Show answer

A written answer, not marked automatically.

Question 16 on its own page

Question 17

+2 marksOne correct option

The probability density function of a continuous random variable is given by

Based on the above data, answer the given subquestions.

Find the expected value of .

A
B
C
D

Show answer

Correct answer

D

Question 17 on its own page

Question 18

+2 marksOne correct option

The joint PMF of two discrete random variables and is given in the following table:

Based on the above data, answer the given subquestions.

What is the value of ?

A
0.01
B
0.10
C
0.70
D
0.30

Show answer

Correct answer

B
0.10

Question 18 on its own page

Question 19

+3 marksWritten answer

The joint PMF of two discrete random variables and is given in the following table:

Based on the above data, answer the given subquestions.

Find the value of . Enter the answer correct to two decimal places.

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Question 19 on its own page

Question 20

+3 marksWritten answer

Let the random variable represent the number of issues resolved during a customer support call. The probability mass function of is given by

where

. A random sample of five customer ratings is {1, 2, 2, 3, 2}. Based on the above data, answer the given subquestions.

Let the random variable represent the number of issues resolved during a customer support call. The probability mass fu

Find the method of moments estimate of for the given sample. Enter the answer correct to two decimal places.

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Question 20 on its own page

Question 21

+2 marksOne correct option

Let the random variable represent the number of issues resolved during a customer support call. The probability mass function of is given by

where

. A random sample of five customer ratings is {1, 2, 2, 3, 2}. Based on the above data, answer the given subquestions.

The service policy requires that the average number of issues resolved per call must be at least 1.5. Based on the estimated value of , determine whether this requirement is satisfied.

A
The requirement is satisfied.
B
The requirement is not satisfied.

Show answer

Correct answer

A
The requirement is satisfied.

Question 21 on its own page

Question 22

+1 markWritten answer

Let be a discrete random variable with the probability mass function

where

is an unknown parameter. Let be an estimator of from i.i.d , given as

Check whether is an unbiased estimator of , otherwise find an unbiased estimator of . Solution: Since we know that will be an unbiased estimator of if . Now,

Now, the values of and can be calculated as Now, substituting the values in equation(1), we get Hence, from equation(2) we can conclude that is an estimator of . Also, will be an unbiased estimator of . Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Show answer

A written answer, not marked automatically.

Question 22 on its own page

Question 23

+1 markWritten answer

A laboratory is testing a new temperature sensor to check whether it provides more consistent readings than the old model. The lab records the temperature (in oC) from 5 repeated measurements using the new sensor: 10, 8, 2, 5, 15 The old temperature sensor has a known standard deviation of . Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 23 on its own page

Question 24

+1 markWritten answer

A factory manufactures light bulbs, and each bulb has a probability of being defective. To estimate this probability using Bayesian methods, assume the prior distribution of to be Beta(3, 3). A quality inspector tests 12 independent bulbs. If 5 bulbs are found to be defective, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 24 on its own page

Question 25

+1 markWritten answer

The probability density function of a continuous random variable is given by

Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 25 on its own page

Question 26

+1 markWritten answer

The joint PMF of two discrete random variables and is given in the following table:

Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 26 on its own page

Question 27

+1 markWritten answer

Let the random variable represent the number of issues resolved during a customer support call. The probability mass function of is given by

where

. A random sample of five customer ratings is {1, 2, 2, 3, 2}. Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 27 on its own page