IITM BS Week 7 Statistics 2 Questions

01

Suppose

X_1, X_2, \dots, X_{100}

are i.i.d random variables with

E[X_i] = \mu

and

Var(X_i) = \sigma^2

for

i=1,2,\dots,100

. Let

\hat{\mu} = \frac{1}{90}\sum_{i=11}^{100}X_i

be an estimator of

\mu

. Find the mean squared error (MSE) of

\hat{\mu}

.

Single correct

MEDIUM

3 marks

10 May 2026

02

The time (in minutes) passengers spend in the airport security line during a quiet period follows a continuous uniform distribution

[4, \theta]

. The waiting times (in minutes) from a random sample of 5 passengers are as follows:

4, 6, 7, 5, 8

. Find the method of moments estimate of

\theta

.

Numerical

MEDIUM

3 marks

10 May 2026

03

Find the method of moments estimate of

\theta

for the given sample. Enter the answer correct to one decimal place.

Comprehension

MEDIUM

3 marks

10 May 2026

04

The experiment is considered efficient if the expected number of trials required to achieve the outcome is at most 4. Using the estimated value of

\theta

, determine whether this requirement is satisfied.

Comprehension

EASY

2 marks

10 May 2026

05

Is

\hat{\mu}

an unbiased estimator of

\mu

?

Comprehension

EASY

2 marks

10 May 2026

06

Consider the derivation for

Var(\hat{\mu})

:\n

Var(\hat{\mu}) = Var(a\bar{X}_1 + b\bar{X}_2)

\nSince the two samples are independent,\n

\implies Var(\hat{\mu}) = \mathbf{A} Var(\bar{X}_1) + \mathbf{B} Var(\bar{X}_2) \quad \dots(1)

\nWhere,\n

Var(\bar{X}_1) = \frac{\mathbf{C}}{n_1}

\n

Var(\bar{X}_2) = \frac{\sigma^2}{n_2}

\nNow, from equation (1):\n

Var(\hat{\mu}) = \sigma^2 \left( \frac{\mathbf{D}}{n_1} + \frac{\mathbf{B}}{n_2} \right)

\nOn substituting the values of

n_1, n_2, \sigma^2, a

and using the relation

a+b=1

, we get:\n

Var(\hat{\mu}) = \mathbf{E}

\n\n**Options list for the blanks:**\n(1)

b

\n(2)

\sigma^2

\n(3)

a

\n(4)

a^2

\n(5)

b^2

\n(6)

a^2

(or is it

\frac{a^2}{n_1}

? No, Option 6 is

a^2

)\n(7)

\frac{a}{n_1}

\n(8)

\frac{\sigma^2}{n_2}

\n(9)

\frac{b^2}{n_2}

\n(10)

\frac{\sigma^2}{n_1^2}

\n\nEnter the correct option number for

\mathbf{A}

.

Comprehension

MEDIUM

0.5 marks

10 May 2026

07

Enter the correct option number for

\mathbf{B}

.

Comprehension

MEDIUM

0.5 marks

10 May 2026

08

Enter the correct option number for

\mathbf{C}

.

Comprehension

MEDIUM

0.5 marks

10 May 2026

09

Enter the correct option number for

\mathbf{D}

.

Comprehension

MEDIUM

0.5 marks

10 May 2026

10

Calculate the value for

\mathbf{E}

. Enter the answer correct to two decimal places.

Comprehension

MEDIUM

1 marks

10 May 2026

11

Suppose

X_1, X_2, \dots, X_n \sim \text{i.i.d. Uniform}(0, \theta)

, and let

\hat{\theta} = \frac{k}{n}\sum_{i=1}^n X_i

be an estimator of

\theta

. Which of the following option(s) is(are) correct?

Multiple correct

MEDIUM

3 marks

13 April 2025

12

Find the method of moments estimate of

\theta

for the given sample.

Comprehension

HARD

4 marks

13 April 2025

13

The company has a policy requirement that the average waiting time must not exceed 3 minutes. Based on the estimated value of

\theta

, determine whether the company meets this requirement.

Comprehension

EASY

1 marks

13 April 2025

14

Find the maximum likelihood estimate of

\lambda

for the given sample. Enter the answer correct to one decimal place.

Comprehension

EASY

3 marks

13 April 2025

15

A four-sided die is such that the probability of getting any number from one to three is

\theta

, and the probability of getting four is

1 - 3\theta

. The die is rolled seven times and outcomes are

1, 4, 3, 4, 3, 2, 4

. Find the estimate of

\theta

using method of moments. Enter the answer correct to two decimal places.

Numerical

MEDIUM

3 marks

22 December 2024

16

Suppose

X_1, X_2, \dots, X_{30}

are i.i.d. samples from a binomial distribution with an unknown parameter

p

and a known parameter

n

. Let

\hat{p}_1, \hat{p}_2

and

\hat{p}_3

be three estimators of

p

given as:

\hat{p}_1 = \frac{\sum_{i=1}^{30} X_i}{30}

\hat{p}_2 = \frac{n \sum_{i=1}^{30} X_i}{30}

\hat{p}_3 = \frac{\sum_{i=1}^{30} X_i}{30n}

Which of the following option(s) is(are) true?

Multiple correct

MEDIUM

3 marks

22 December 2024

17

Consider a random sample

(1, 0, 1, 2, 2, 0, 1)

. Find the method of moments estimate of

\theta

for the given sample. Enter the answer correct to two decimal places.

Comprehension

MEDIUM

2 marks

22 December 2024

18

Consider a random sample

(2, 0, 1, 1, 2, 2, 1, 1, 0, 0)

. Find the maximum likelihood estimate of

\theta

for the given sample. Enter the answer correct to one decimal place.

Comprehension

MEDIUM

3 marks

22 December 2024

19

Consider a group of individuals participating in a fitness program, where the weights they can lift follow a normal distribution with an unknown mean

\mu

and standard deviation

\sigma

. A sample of

9

participants was randomly selected, and their lifting capacities (in kgs) are as follows: 60, 66, 40, 54, 40, 60, 48, 60, 52 Find the maximum likelihood estimate of

\mu

.

Numerical

EASY

3 marks

01 September 2024

20

Let

X_1, X_2, \ldots, X_n \sim \text{iid Exp}(\lambda)

, and let

\bar{X}

and

S^2

be the sample mean and sample variance, respectively. Select the correct options from the following:

Single correct

MEDIUM

3 marks

01 September 2024

21

Find the method of moments estimator of

\theta

.

Comprehension

MEDIUM

2 marks

01 September 2024

22

Is the previous question estimator obtained unbiased?

Comprehension

EASY

1 marks

01 September 2024

23

Suppose the time taken by a delivery service to deliver a package (in hours) follows an exponential distribution with parameter

\lambda

. The timings (in hours) from a random sample of 6 deliveries are as follows:

15, 20, 15, 16, 14, 20

Find the method of moment estimate of

\lambda

for the given sample. Enter the answer correct to two decimal places.

Numerical

EASY

3 marks

28 April 2024

24

Suppose

X_1, X_2, \dots, X_n

are i.i.d. samples from a distribution

X

with an unknown mean

\mu

and variance

\sigma^2

. If

\hat{\mu}_1

is an estimator of

\mu

such that

E[\hat{\mu}_1] = 2\mu + 3

and another estimator

\hat{\mu}_2

is defined as

\hat{\mu}_2 = \frac{\hat{\mu}_1 - k}{k-1}

Find the value of

k

for which

\hat{\mu}_2

will be an unbiased estimator of

\mu

.

Numerical

MEDIUM

3 marks

28 April 2024

25

Consider a random sample

(0, 3, 2, 2, 3, 1, 2, 2, 3, 0)

. Find the method of moments estimate of

p

for the given sample. Enter the answer correct to one decimal place.

Comprehension

MEDIUM

3 marks

28 April 2024

26

Consider a random sample

(0, 0, 1, 2, 2, 3, 3, 3, 3, 3)

. Find the maximum likelihood estimate of

p

for the given sample. Enter the answer correct to two decimal places.

Comprehension

MEDIUM

2 marks

28 April 2024

27

The time (in minutes) that buses will reach on a particular stand inside IIT Madras campus follows the continuous uniform distribution

[10, \theta]

. The timings (in minutes) from a random sample of 5 buses are as follows:

10, 12, 15, 13, 10

Find the estimate of

\theta

by using method of moments.

Numerical

MEDIUM

3 marks

24 December 2023

28

Suppose

X_1, X_2, \dots, X_n \sim \text{iid Normal}(\mu, \sigma^2)

with sample mean,

\bar{X} = \frac{\sum_{i=1}^n X_i}{n}

and sample variance,

S^2 = \frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n - 1}

. Choose the correct option(s) from the following.

Multiple correct

MEDIUM

3 marks

24 December 2023

29

Let

X_1, X_2, \dots, X_n

be i.i.d. samples from a distribution with an unknown mean

\mu

and standard deviation 4. Let

\hat{\mu} = \frac{X_1 + X_4}{2}

be an estimator of

\mu

. Which of the following option(s) is(are) correct?

Multiple correct

MEDIUM

3 marks

24 December 2023

30

Let a random sample of

X

be

1, 2, 2.5, 2.5, 2

. Find the method of moments estimate of

\alpha

for the given sample.

Comprehension

MEDIUM

2 marks

24 December 2023

31

Let a random sample of

X

be

6, 7.5, 7.5, 3, 6

. Find the maximum likelihood estimate of

\alpha

for the given sample.

Comprehension

MEDIUM

3 marks

24 December 2023

32

Let

X_1, X_2, \dots, X_{100} \sim \text{Uniform}(0, \theta)

. Which of the following is an unbiased estimator for

\theta

?

Single correct

EASY

3 marks

03 September 2023

33

Find the method of moments estimate of

p

. Enter the answer correct to two decimal places.

Comprehension

MEDIUM

2 marks

03 September 2023

34

Find the maximum likelihood estimate of

p

. Enter the answer correct to one decimal place.

Comprehension

MEDIUM

3 marks

03 September 2023

35

Suppose

X \sim \text{Uniform}[0, \theta]

, where

\theta

is an unknown constant. Consider a sample

(4, 7, 8, 11, 5)

. Let

\hat{\theta}_{MME}

and

\hat{\theta}_{ML}

be the method of moments estimator and the maximum likelihood estimator of

\theta

, respectively. Find

|\hat{\theta}_{MME} - \hat{\theta}_{ML}|

.

Numerical

MEDIUM

3 marks

30 April 2023

36

Suppose

X_1, X_2, X_3

are i.i.d. samples from a distribution

X

with an unknown mean

\mu

and variance

\sigma^2

. Let

\hat{\mu}_1 = X_1 + X_2 - X_3

,

\hat{\mu}_2 = 3X_1 + 2X_2 - 4X_3

and

\hat{\mu}_3 = \frac{X_1}{2} + X_2 - \frac{X_3}{2}

be three unbiased estimators of

\mu

. Which of the following is true?

Single correct

MEDIUM

3 marks

30 April 2023

37

Consider a random sample

(-0.2, 0.3, 0.7, -0.6, 0.1)

. Find the method of moments estimate of

\theta

for the given sample. Enter the answer correct to two decimal places.

Comprehension

MEDIUM

2 marks

30 April 2023

38

Consider a random sample

\left(-\frac{1}{2}, 0, \frac{1}{3}, 0, \frac{1}{3}\right)

. Find the maximum likelihood estimate of

\theta

for the given sample. Enter the answer correct to two decimal places.

Comprehension

HARD

3 marks

30 April 2023

39

The heights (in cm) of people in a certain area are normally distributed with mean

\mu

and standard deviation 40. The heights from a random sample of 10 people are as follows:

155, 159, 145, 155, 150, 160, 165, 157, 169, 145

Suppose

\hat{\mu}

is the maximum likelihood estimate for

\mu

. Letting

X \sim Normal(\hat{\mu}, 2)

, find the value of

f_X(158)

. Enter the answer correct to two decimal places.

Numerical

MEDIUM

3 marks

11 December 2022

40

Consider an estimator

\hat{\theta} = \frac{2}{n}\sum_{i=1}^n X_i

of

\theta

. Is

\hat{\theta}

an unbiased estimator of

\theta

?

Comprehension

EASY

1 marks

11 December 2022

41

Assume

\theta

is an unknown parameter. Find the maximum likelihood estimate of

\theta

.

Comprehension

MEDIUM

2 marks

11 December 2022

42

Consider

\theta

as a random variable with prior distribution as Uniform [0,1]. Choose the correct options from the following:

Comprehension

MEDIUM

2 marks

11 December 2022

43

Find the method of moments estimate of

\theta

for the given sample. Enter the answer correct to one decimal place.

Comprehension

MEDIUM

2 marks

11 December 2022

44

Find the maximum likelihood estimate of

\theta

for the given sample. Enter the answer correct to one decimal place.

Comprehension

HARD

3 marks

11 December 2022

45

Let

X_1, X_2, \ldots, X_n \sim \text{i.i.d. Uniform}[-a, a]

, where

a

is unknown. An estimator

\hat{a}

for

a

is given as:

\hat{a} = \frac{3}{n}(X_1^2 + X_2^2 + \ldots + X_n^2)

Is

\hat{a}

an unbiased estimator of

a

?

Single correct

MEDIUM

2 marks

07 August 2022

46

The average life of a battery is 4 years, with a standard deviation of 1 year. Assuming that the lives of the batteries follow approximately a normal distribution, find the probability that the mean life of a random sample of 30 such batteries falls between 3.8 and 4.2 years. Enter your answer correct to two decimal places.

Numerical

MEDIUM

4 marks

07 August 2022

47

Find the method of moments estimate of

\lambda

for the given sample. Enter the answer correct to three decimal places.

Comprehension

EASY

2 marks

07 August 2022

48

Find the maximum likelihood estimate of

\lambda

for the given sample. Enter the answer correct to three decimal places.

Comprehension

EASY

2 marks

07 August 2022

All PYQs

All PYQs