Week 3 Statistics 2 Questions

Q1

A random variable

X

has the moment generating function

M_X(\lambda)=\dfrac{e^{\lambda/10}-1}{\lambda/10},\quad \lambda\ne 0.

Find the variance of

X

. Enter the answer correct to three decimal places.

Integer answermedium3 marksAPRIL 06, 2026

APRIL 06, 2026

Q2

Using Chebyshev's inequality, find the smallest integer value

n

such that

P(|\bar{X}_n-\mu|\ge 0.1)\le 0.2

, where

\sigma^2=1

.

Integer answermedium3 marksAPRIL 06, 2026

APRIL 06, 2026

Q3

Suppose

X_1,X_2,X_3\sim

i.i.d.

X

such that

E[X]=10

and

\operatorname{Var}(X)=4

. Define

Y=X_1-X_2+X_3

and

Z=2Y

. Choose the correct option(s).

Multiple correcteasy3 marksAUGUST 03, 2025

AUGUST 03, 2025

Q4

At a customer support center, the number of service requests received in a day is modeled by a Poisson random variable

X

with mean

\lambda=18

. The total time spent processing requests and total words written are modeled as

T=7X+12

and

W=4X-1

. Find

\operatorname{Cov}(T,W)

.

Integer answereasy3 marksAUGUST 03, 2025

AUGUST 03, 2025

Q5

If

Y=\bar X_1-\bar X_2

, use Chebyshev’s inequality to find a lower bound for

P(|Y|<18)

. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksAUGUST 03, 2025

AUGUST 03, 2025

Q6

Let

X\sim\mathrm{Bernoulli}(0.3)

. Find the MGF of the centered version of

X

.

Single correcteasy3 marksMARCH 16, 2025

MARCH 16, 2025

Q7

Using Chebyshev’s inequality, find a lower bound for

P(|Y-0.5|<0.6)

. Enter the answer correct to three decimal places.

Comprehensionmedium3 marksMARCH 16, 2025

MARCH 16, 2025

Q8

Find

E\left[Y\right]

for

Y=\sum_{i=1}^{4}X_i+2\sum_{i=16}^{25}X_i

.

Comprehensionmedium2 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q9

Using Chebyshev’s inequality, find an upper bound for

P(|\bar X-0.5|>0.25)

, where

\bar X=(X_1+\cdots+X_{16})/16

. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q10

If

\operatorname{Cov}(X,Y)=0

, find

E[XY]

.

Comprehensionmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q11

Which option(s) is/are correct?

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q12

Let

X_1

and

X_2

be i.i.d., where

X

has PMF

P(X=0)=0.2

,

P(X=1)=0.4

,

P(X=2)=0.4

. Define

Y=X_1+X_2

. Find the MGF of

Y

.

Single correcteasy3 marksAUGUST 04, 2024

AUGUST 04, 2024

Q13

Consider a random variable

X

with

E[X]=1

,

E[X^2]=0

, and

E[X^3]=2

. Define

Y=-1+X+3X^2

. Find

\operatorname{Cov}(X,Y)

.

Integer answermedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

Q14

Using Chebyshev’s inequality, find the least upper bound on the probability that the sample mean deviates from the population mean by more than

0.5

hours.

Comprehensionmedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

Q15

Suppose

X

and

Y

are two independent random variables with probability density functions

f(x)=\frac{8}{x^3}

for

x>2

and

0

otherwise, and

g(y)=2y

for

0<y<1

and

0

otherwise. Calculate the value of

E[XY]

.

Single correctmedium3 marksMARCH 24, 2024

MARCH 24, 2024

Q16

Which of the following inequalities result from Chebyshev's inequality?

Comprehensionmedium3 marksMARCH 24, 2024

MARCH 24, 2024

Q17

Find the minimum value of

n

such that the sample mean lies in

[\mu-5,\mu+5]

with probability more than

0.95

using Chebyshev's inequality.

Comprehensionmedium2 marksMARCH 24, 2024

MARCH 24, 2024

Q18

Find the value of

\mathrm{Cov}(X,Y)

.

Comprehensionmedium3 marksMARCH 24, 2024

MARCH 24, 2024

Q19

What conclusion will you make based on the obtained value in the given part?

Comprehensioneasy2 marksMARCH 24, 2024

MARCH 24, 2024

Q20

Which of the following inequalities are true with respect to Chebyshev inequality?

Comprehensionmedium3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Q21

Find the minimum value of

n

such that the sample mean lies in

[2.5,3.5]

with probability more than

0.95

using Chebyshev inequality.

Comprehensionmedium3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Q22

Suppose

X

is a discrete random variable and has moment generating function

M_X(t)=\frac{1}{7}+\frac{3}{7}e^{2t}+\frac{2}{7}e^{4t}+\frac{1}{7}e^{6t}

. What is the PMF of

X

?

Single correcteasy3 marksAUGUST 06, 2023

AUGUST 06, 2023

Q23

Suppose

X_1,X_2,X_3,X_4\sim \text{i.i.d. }X

with

E[X]=10

and

\operatorname{Var}(X)=4

. Define

S=2X_1-2X_2-X_3+3X_4

. Choose the correct option(s) from below:

Multiple correctmedium3 marksAUGUST 06, 2023

AUGUST 06, 2023

Q24

Find the expected value of

Y=\sum_{i=1}^{20}X_i+\sum_{i=11}^{20}X_i

.

Comprehensioneasy2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q25

Using Chebyshev's inequality, find an upper bound for

P(|\bar{X}-10|>2)

, where

\bar{X}=(X_1+X_2+\cdots+X_{20})/20

is the sample mean. Enter the answer correct to 3 decimal places.

Comprehensionmedium3 marksAUGUST 06, 2023

AUGUST 06, 2023

Q26

Let

X_1,X_2,\ldots,X_n

be i.i.d.

X

with mean

\mu=0

and variance

\sigma^2=1

. Using Chebyshev's inequality, what should be the minimum value of

n

such that the probability that the sample mean

\frac{X_1+X_2+\cdots+X_n}{n}

lies between

-0.5

and

0.5

is at least

0.95

?

Single correctmedium3 marksAPRIL 02, 2023

APRIL 02, 2023

Q27

Suppose

X_1,X_2,X_3,X_4

are i.i.d. Bernoulli

(2/3)

. Define

Y=2X_1+3X_2+4X_3+5X_4

. Find

\mathrm{Var}(Y)

.

Integer answermedium3 marksAPRIL 02, 2023

APRIL 02, 2023

Q28

Let

X_1,X_2,\ldots,X_n

be i.i.d. Poisson

(9)

. Using Chebyshev's inequality, what should be the minimum value of

n

such that the probability that the sample mean

\bar X

lies between

8.6

and

9.4

is at least

0.95

?

Integer answermedium3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q29

Find the moment generating function of the random variable

Y

.

Comprehensionmedium3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q30

Find the expected value of

Y

. Enter the answer correct to two decimal places.

Comprehensioneasy2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Statistics 2 - Week 3