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Statistics 2 - Week 5

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SubjectStatistics 2

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Q1-Q1 of 83

The joint PDF of two continuous random variables

X

Y

f_{XY}(x,y)=\begin{cases}\dfrac{4xy}{17^4}, & 0\le x\le 17,\ 0\le y\le 17,\\ 0, & \text{otherwise.}\end{cases}

X

Y

Single correctmedium3 marksAPRIL 06, 2026

X_1, X_2, X_3, X_4

be i.i.d. Normal random variables. Find the value of

P(3X_1+X_2+2X_3+3X_4\le 17)

Single correctmedium3 marksAPRIL 06, 2026

Enter the correct option number for __A__.

Comprehensioneasy0.5 marksAPRIL 06, 2026

Enter the correct option number for __B__.

Comprehensioneasy0.5 marksAPRIL 06, 2026

Enter the correct option number for __C__.

Comprehensioneasy0.5 marksAPRIL 06, 2026

Enter the correct option number for __D__.

Comprehensioneasy0.5 marksAPRIL 06, 2026

Enter the correct option number for __E__.

Comprehensioneasy0.5 marksAPRIL 06, 2026

(X,Y)\sim \text{Uniform}(R)

R=\{(x,y):x^2+y^2<9,\ x>0,\ y>0\}

. Find the value of

P(X<1,Y<2)

. Enter the answer correct to two decimal places.

Integer answermedium2 marksAPRIL 06, 2026

Which of the following statements is/are correct?

Comprehensionmedium2 marksAPRIL 06, 2026

What is the marginal PDF of

X

x=50

Comprehensionmedium3 marksAPRIL 06, 2026

A randomly selected patient from this age group is found to have an enzyme level of

50

U/L. What is the probability that the patient is healthy?

Comprehensionmedium2 marksAPRIL 06, 2026

Which of the following statements is correct about the distribution of the sample mean?

Comprehensioneasy2 marksAPRIL 06, 2026

Find the probability that the sample mean lifetime exceeds

506

hours. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksAPRIL 06, 2026

X

Y

be continuous random variables with joint PDF

f_{X,Y}(x,y)=\frac{2x}{9}

0\le x\le2

1\le y\le3

0

otherwise. Find

P(X<1\mid Y=2)

correct to two decimal places.

Integer answermedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

What is the marginal density function

f_X(x)

Comprehensionmedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

P(0<X<1,\ Y<1)

. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksAUGUST 03, 2025

AUGUST 03, 2025

What is the marginal distribution of

Y

Comprehensionmedium2 marksAUGUST 03, 2025

AUGUST 03, 2025

f_{X|Y=0.4}(1)

. Enter the answer correct to two decimal places.

Comprehensionhard3 marksAUGUST 03, 2025

AUGUST 03, 2025

Which option is an acceptable approximate model for

Y

as per the Central Limit Theorem?

Comprehensionmedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

Using the Central Limit Theorem, it is calculated that

P(Y>a)\approx0.0228

a

? Use the useful standard normal data given in the question.

Comprehensionmedium2 marksAUGUST 03, 2025

AUGUST 03, 2025

(X,Y)\sim\mathrm{Uniform}(D)

D=\{(x,y):x+y<2,x>0,y>0\}

. What is the marginal density function of

X

Single correctmedium3 marksMARCH 16, 2025

Find the value of

k

Comprehensioneasy2 marksMARCH 16, 2025

P(X<1/2,Y<1/2)

Comprehensionmedium3 marksMARCH 16, 2025

Which option(s) is/are correct?

Comprehensioneasy2 marksMARCH 16, 2025

Which option is an acceptable approximate model for

S

as per the Central Limit Theorem?

Comprehensioneasy2 marksMARCH 16, 2025

Using CLT, find an approximate probability that

S

40

48

heads. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksMARCH 16, 2025

X_1,X_2,\ldots,X_{10}

be i.i.d. random variables with common MGF

M_X(\lambda)

. Find the MGF of

\bar X

\bar X=(X_1+\cdots+X_{10})/10

Single correctmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

X

be study hours and

Y

recreation hours. The joint PDF is

f_{XY}(x,y)=c(x+y)

0\le x\le1

0\le y\le2

0

otherwise. Find

21c

Integer answereasy3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

X

represent the total service time for

100

customers. Which option is an acceptable approximate model for

X

as per the CLT?

Comprehensioneasy2 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Using CLT, find the approximate probability that the total service time exceeds

1200

minutes. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Find the joint density function

f_{XY}(x,y)

Comprehensioneasy3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

P(X+Y<2)

. Enter the answer correct to one decimal place.

Comprehensionmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

A survey estimates the proportion of voters supporting a candidate. The true proportion is

0.45

and a random sample of

500

voters is taken. Using the Central Limit Theorem, what is the approximate probability that the sample proportion is between

0.43

0.47

Single correctmedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

If the joint PDF is

f_{XY}(x,y)=1/k

(x,y)\in D

0

otherwise, find

k

correct to two decimal places.

Comprehensioneasy1 marksAUGUST 04, 2024

AUGUST 04, 2024

Which option(s) are true?

Comprehensionmedium2 marksAUGUST 04, 2024

AUGUST 04, 2024

Find the conditional distribution

f_{Y|X}(y|x)

Comprehensionmedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

P(X<1/2\mid Y=1/2)

Comprehensioneasy2 marksAUGUST 04, 2024

AUGUST 04, 2024

What is the expected value of the sample mean time spent on social media?

Comprehensioneasy2 marksAUGUST 04, 2024

AUGUST 04, 2024

Using CLT, what is the approximate distribution of the sample mean recovery time?

Comprehensioneasy2 marksAUGUST 04, 2024

AUGUST 04, 2024

Using CLT, find the approximate probability that the sample mean recovery time is between

4

6

days. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

What is the distribution of

X

Comprehensionmedium3 marksAUGUST 04, 2024

AUGUST 04, 2024

P(W=1\mid X=-1)

. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksAUGUST 04, 2024

AUGUST 04, 2024

X_1, X_2, \ldots, X_{30}

be i.i.d. Bernoulli

(0.8)

. Define a new random variable

Y=X_1+X_2+\cdots+X_{30}

. Using the Central Limit Theorem, find the approximate value of

P(Y>21)

. Enter the answer correct to two decimal places.

Integer answermedium3 marksMARCH 24, 2024

X

Y

be two continuous random variables with joint PDF

f_{XY}(x,y)=x+cy^2

0\le x\le 1

0\le y\le 1

0

otherwise. Find the value of

c

. Enter the answer correct to one decimal place.

Integer answereasy3 marksMARCH 24, 2024

X_1, X_2, \ldots, X_{100}

be i.i.d. samples with mean

150

100

. Find the mean and variance of

\bar{X}

\bar{X}=\frac{X_1+X_2+\cdots+X_{100}}{100}

is the sample mean.

Single correcteasy3 marksMARCH 24, 2024

Find the marginal density of

X

Comprehensionmedium3 marksMARCH 24, 2024

What is the value of

E[X]

? Enter the answer correct to one decimal place.

Comprehensioneasy2 marksMARCH 24, 2024

Let a random variable

X

denote the total number of students without laptops in the selected sample. Which of the following options is a good model for

X

Comprehensioneasy2 marksMARCH 24, 2024

Using the Central Limit Theorem, find an approximate probability that at least

80

students in the selected random sample own a laptop. Enter the answer correct to one decimal place.

Comprehensionmedium3 marksMARCH 24, 2024

Find the value of

k

Comprehensioneasy2 marksMARCH 24, 2024

What is the value of

P(X<\frac12, Y<\frac12)

Comprehensionmedium3 marksMARCH 24, 2024

(X,Y)\sim \mathrm{Uniform}(D)

D=\{(x,y):x+y<3,\ x>0,\ y>0\}

. What is the marginal density function of

f_X(x)

Single correctmedium3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Suppose the moment generating function of a random variable

X

M_X(t)=e^{2t^2}

X_1,X_2,X_3,X_4

be i.i.d. samples with distribution

X

Y=X_1+X_2+X_3+X_4

. Which of the following options is true?

Single correcteasy3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

X

Y

Comprehensioneasy2 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Find the value of

P(X<Y)

. Enter the answer correct to one decimal place.

Comprehensioneasy3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Using Central Limit theorem, find the approximate probability that there are more than

53

errors in a certain data file. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksDECEMBER 03, 2024

DECEMBER 03, 2024

Find the joint density function

f_{XY}(x,y)

Comprehensioneasy1 marksAUGUST 06, 2023

AUGUST 06, 2023

P(|X-Y|<1)

. Enter the answer correct to two decimal places.

Comprehensionmedium4 marksAUGUST 06, 2023

AUGUST 06, 2023

Find the value of

c

. Enter the answer correct to two decimal places.

Comprehensioneasy2 marksAUGUST 06, 2023

AUGUST 06, 2023

What is the probability that the pressure change during the chemical process is more than

0.5

, given that the temperature change is equal to

0.5

? Enter the answer correct to two decimal places.

Comprehensionmedium3 marksAUGUST 06, 2023

AUGUST 06, 2023

Let a random variable

X

denote the total number of children who have normal eye-sight in the selected sample. Which of the following is true?

Comprehensioneasy2 marksAUGUST 06, 2023

AUGUST 06, 2023

Using the Central Limit Theorem, find the approximate probability that in a random sample of

200

selected children at least

20

will have defective eye-sight. Enter the answer correct to 1 decimal place.

Comprehensionmedium3 marksAUGUST 06, 2023

AUGUST 06, 2023

(X,Y)\sim \mathrm{Uniform}(D)

D=\{(x,y):2<x+y<4,\ x>0,\ y>0\}

P(Y<2X)

Single correctmedium3 marksAPRIL 02, 2023

Find the PDF of the marks of a candidate chosen uniformly at random.

Comprehensionmedium3 marksAPRIL 02, 2023

If a randomly selected candidate got

60

marks, what is the probability that the selected candidate is a boy?

Comprehensionmedium2 marksAPRIL 02, 2023

Find the value of

c

. Enter the answer correct to one decimal place.

Comprehensionmedium2 marksAPRIL 02, 2023

Find the probability that the amount of petrol sold in a given week is less than half the amount stocked in that week. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksAPRIL 02, 2023

Suppose a sample of

20

bottles is selected for a quality inspection. Let

X

denote the total number of bottles that are of bad quality in the selected sample. Which of the following is true?

Comprehensioneasy2 marksAPRIL 02, 2023

Using the Central Limit Theorem, find the approximate probability that a machine will produce more than

370

bottles that are of good quality on a particular day. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksAPRIL 02, 2023

Which of the following statements are true?

Comprehensionmedium3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

X

Y

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Suppose iron rods are manufactured with a mean weight of

10

kg and a standard deviation of

0.4

kg. How does the variance of the sample mean change when the sample size is increased from

25

64

Single correcteasy3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Y

represents the total number of bits without an error in a certain data file. Which of the following is true?

Comprehensioneasy2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Using the Central Limit Theorem, find the approximate probability that there are more than

45

errors in a certain data file. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Find the value of

c

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Find the probability that the stock price of A will be higher than that of B. Enter the answer correct to two decimal places.

Comprehensionmedium3 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

X_1,X_2,\ldots,X_{30}

be i.i.d. Poisson

(2)

Y=\sum_{i=1}^{30}X_i

. Using the Central Limit Theorem, find the value of

P(Y>20)

Single correctmedium3 marksJULY 10, 2022

Suppose random variables

X

Y

are uniformly distributed over the region

D=([-1,0]\times[0,1])\cup([0,2]\times[-2,0])

. Choose the correct options from the following.

Multiple correctmedium3 marksJULY 10, 2022

P(0<X<1/2,\ 1/4<Y<1/2)

. Enter the answer correct to three decimal places.

Comprehensioneasy1 marksJULY 10, 2022

P(0<X<1/2)

. Enter the answer correct to one decimal place.

Comprehensioneasy1 marksJULY 10, 2022

P(X<2Y)

. Enter the answer correct to two decimal places.

Comprehensionmedium2 marksJULY 10, 2022

Find the minimum value of

n

such that the probability that the sample mean

\bar X

0.2

of the distribution mean is at least

0.9

using the Weak Law of Large Numbers.

Comprehensionmedium3 marksJULY 10, 2022

Find the distribution of runs of a randomly chosen player.

Comprehensionmedium3 marksJULY 10, 2022

Showing 83 questions.