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Week 5 Statistics 2 Questions | Prasnya
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83Q
01
The joint PDF of two continuous random variables
X
X
X
and
Y
Y
Y
is given by
f
X
Y
(
x
,
y
)
=
{
4
x
y
17
4
,
0
≤
x
≤
17
,
Â
0
≤
y
≤
17
,
0
,
otherwise.
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}
f
X
Y
​
(
x
,
y
)
=
⎩
⎨
⎧
​
1
7
4
4
x
y
​
,
0
,
​
0
≤
x
≤
17
,
Â
0
≤
y
≤
17
,
otherwise.
​
Are
X
X
X
and
Y
Y
Y
independent?
Single correct
MEDIUM
3 marks
06 April 2026
02
Let
X
1
,
X
2
,
X
3
,
X
4
X_1, X_2, X_3, X_4
X
1
​
,
X
2
​
,
X
3
​
,
X
4
​
be i.i.d. Normal random variables. Find the value of
P
(
3
X
1
+
X
2
+
2
X
3
+
3
X
4
≤
17
)
P(3X_1+X_2+2X_3+3X_4\le 17)
P
(
3
X
1
​
+
X
2
​
+
2
X
3
​
+
3
X
4
​
≤
17
)
.
Single correct
MEDIUM
3 marks
06 April 2026
03
Enter the correct option number for __A__.
Comprehension
EASY
0.5 marks
06 April 2026
04
Enter the correct option number for __B__.
Comprehension
EASY
0.5 marks
06 April 2026
05
Enter the correct option number for __C__.
Comprehension
EASY
0.5 marks
06 April 2026
06
Enter the correct option number for __D__.
Comprehension
EASY
0.5 marks
06 April 2026
07
Enter the correct option number for __E__.
Comprehension
EASY
0.5 marks
06 April 2026
08
Let
(
X
,
Y
)
∼
Uniform
(
R
)
(X,Y)\sim \text{Uniform}(R)
(
X
,
Y
)
∼
Uniform
(
R
)
, where
R
=
{
(
x
,
y
)
:
x
2
+
y
2
<
9
,
Â
x
>
0
,
Â
y
>
0
}
R=\{(x,y):x^2+y^2<9,\ x>0,\ y>0\}
R
=
{(
x
,
y
)
:
x
2
+
y
2
<
9
,
Â
x
>
0
,
Â
y
>
0
}
. Find the value of
P
(
X
<
1
,
Y
<
2
)
P(X<1,Y<2)
P
(
X
<
1
,
Y
<
2
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
2 marks
06 April 2026
09
Which of the following statements is/are correct?
Comprehension
MEDIUM
2 marks
06 April 2026
10
What is the marginal PDF of
X
X
X
at
x
=
50
x=50
x
=
50
?
Comprehension
MEDIUM
3 marks
06 April 2026
11
A randomly selected patient from this age group is found to have an enzyme level of
50
50
50
U/L. What is the probability that the patient is healthy?
Comprehension
MEDIUM
2 marks
06 April 2026
12
Which of the following statements is correct about the distribution of the sample mean?
Comprehension
EASY
2 marks
06 April 2026
13
Find the probability that the sample mean lifetime exceeds
506
506
506
hours. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
06 April 2026
14
Let
X
X
X
and
Y
Y
Y
be continuous random variables with joint PDF
f
X
,
Y
(
x
,
y
)
=
2
x
9
f_{X,Y}(x,y)=\frac{2x}{9}
f
X
,
Y
​
(
x
,
y
)
=
9
2
x
​
,
0
≤
x
≤
2
0\le x\le2
0
≤
x
≤
2
,
1
≤
y
≤
3
1\le y\le3
1
≤
y
≤
3
, and
0
0
0
otherwise. Find
P
(
X
<
1
∣
Y
=
2
)
P(X<1\mid Y=2)
P
(
X
<
1
∣
Y
=
2
)
correct to two decimal places.
Numerical
MEDIUM
3 marks
03 August 2025
15
What is the marginal density function
f
X
(
x
)
f_X(x)
f
X
​
(
x
)
?
Comprehension
MEDIUM
3 marks
03 August 2025
16
Find
P
(
0
<
X
<
1
,
Â
Y
<
1
)
P(0<X<1,\ Y<1)
P
(
0
<
X
<
1
,
Â
Y
<
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
03 August 2025
17
What is the marginal distribution of
Y
Y
Y
?
Comprehension
MEDIUM
2 marks
03 August 2025
18
Find
f
X
∣
Y
=
0.4
(
1
)
f_{X|Y=0.4}(1)
f
X
∣
Y
=
0.4
​
(
1
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
03 August 2025
19
Which option is an acceptable approximate model for
Y
Y
Y
as per the Central Limit Theorem?
Comprehension
MEDIUM
3 marks
03 August 2025
20
Using the Central Limit Theorem, it is calculated that
P
(
Y
>
a
)
≈
0.0228
P(Y>a)\approx0.0228
P
(
Y
>
a
)
≈
0.0228
. What is
a
a
a
? Use the useful standard normal data given in the question.
Comprehension
MEDIUM
2 marks
03 August 2025
21
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim\mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
where
D
=
{
(
x
,
y
)
:
x
+
y
<
2
,
x
>
0
,
y
>
0
}
D=\{(x,y):x+y<2,x>0,y>0\}
D
=
{(
x
,
y
)
:
x
+
y
<
2
,
x
>
0
,
y
>
0
}
. What is the marginal density function of
X
X
X
?
Single correct
MEDIUM
3 marks
16 March 2025
22
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
16 March 2025
23
What is
P
(
X
<
1
/
2
,
Y
<
1
/
2
)
P(X<1/2,Y<1/2)
P
(
X
<
1/2
,
Y
<
1/2
)
?
Comprehension
MEDIUM
3 marks
16 March 2025
24
Which option(s) is/are correct?
Comprehension
EASY
2 marks
16 March 2025
25
Which option is an acceptable approximate model for
S
S
S
as per the Central Limit Theorem?
Comprehension
EASY
2 marks
16 March 2025
26
Using CLT, find an approximate probability that
S
S
S
lies between
40
40
40
and
48
48
48
heads. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
16 March 2025
27
Let
X
1
,
X
2
,
…
,
X
10
X_1,X_2,\ldots,X_{10}
X
1
​
,
X
2
​
,
…
,
X
10
​
be i.i.d. random variables with common MGF
M
X
(
λ
)
M_X(\lambda)
M
X
​
(
λ
)
. Find the MGF of
X
ˉ
\bar X
X
ˉ
, where
X
ˉ
=
(
X
1
+
⋯
+
X
10
)
/
10
\bar X=(X_1+\cdots+X_{10})/10
X
ˉ
=
(
X
1
​
+
⋯
+
X
10
​
)
/10
.
Single correct
MEDIUM
3 marks
01 December 2024
28
Let
X
X
X
be study hours and
Y
Y
Y
recreation hours. The joint PDF is
f
X
Y
(
x
,
y
)
=
c
(
x
+
y
)
f_{XY}(x,y)=c(x+y)
f
X
Y
​
(
x
,
y
)
=
c
(
x
+
y
)
for
0
≤
x
≤
1
0\le x\le1
0
≤
x
≤
1
,
0
≤
y
≤
2
0\le y\le2
0
≤
y
≤
2
, and
0
0
0
otherwise. Find
21
c
21c
21
c
.
Numerical
EASY
3 marks
01 December 2024
29
Let
X
X
X
represent the total service time for
100
100
100
customers. Which option is an acceptable approximate model for
X
X
X
as per the CLT?
Comprehension
EASY
2 marks
01 December 2024
30
Using CLT, find the approximate probability that the total service time exceeds
1200
1200
1200
minutes. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
01 December 2024
31
Find the joint density function
f
X
Y
(
x
,
y
)
f_{XY}(x,y)
f
X
Y
​
(
x
,
y
)
.
Comprehension
EASY
3 marks
01 December 2024
32
Find
P
(
X
+
Y
<
2
)
P(X+Y<2)
P
(
X
+
Y
<
2
)
. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
3 marks
01 December 2024
33
A survey estimates the proportion of voters supporting a candidate. The true proportion is
0.45
0.45
0.45
and a random sample of
500
500
500
voters is taken. Using the Central Limit Theorem, what is the approximate probability that the sample proportion is between
0.43
0.43
0.43
and
0.47
0.47
0.47
?
Single correct
MEDIUM
3 marks
04 August 2024
34
If the joint PDF is
f
X
Y
(
x
,
y
)
=
1
/
k
f_{XY}(x,y)=1/k
f
X
Y
​
(
x
,
y
)
=
1/
k
for
(
x
,
y
)
∈
D
(x,y)\in D
(
x
,
y
)
∈
D
and
0
0
0
otherwise, find
k
k
k
correct to two decimal places.
Comprehension
EASY
1 marks
04 August 2024
35
Which option(s) are true?
Comprehension
MEDIUM
2 marks
04 August 2024
36
Find the conditional distribution
f
Y
∣
X
(
y
∣
x
)
f_{Y|X}(y|x)
f
Y
∣
X
​
(
y
∣
x
)
.
Comprehension
MEDIUM
3 marks
04 August 2024
37
Find
P
(
X
<
1
/
2
∣
Y
=
1
/
2
)
P(X<1/2\mid Y=1/2)
P
(
X
<
1/2
∣
Y
=
1/2
)
.
Comprehension
EASY
2 marks
04 August 2024
38
What is the expected value of the sample mean time spent on social media?
Comprehension
EASY
2 marks
04 August 2024
39
Using CLT, what is the approximate distribution of the sample mean recovery time?
Comprehension
EASY
2 marks
04 August 2024
40
Using CLT, find the approximate probability that the sample mean recovery time is between
4
4
4
and
6
6
6
days. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
04 August 2024
41
What is the distribution of
X
X
X
?
Comprehension
MEDIUM
3 marks
04 August 2024
42
Find
P
(
W
=
1
∣
X
=
−
1
)
P(W=1\mid X=-1)
P
(
W
=
1
∣
X
=
−
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
04 August 2024
43
Let
X
1
,
X
2
,
…
,
X
30
X_1, X_2, \ldots, X_{30}
X
1
​
,
X
2
​
,
…
,
X
30
​
be i.i.d. Bernoulli
(
0.8
)
(0.8)
(
0.8
)
. Define a new random variable
Y
=
X
1
+
X
2
+
⋯
+
X
30
Y=X_1+X_2+\cdots+X_{30}
Y
=
X
1
​
+
X
2
​
+
⋯
+
X
30
​
. Using the Central Limit Theorem, find the approximate value of
P
(
Y
>
21
)
P(Y>21)
P
(
Y
>
21
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
24 March 2024
44
Let
X
X
X
and
Y
Y
Y
be two continuous random variables with joint PDF
f
X
Y
(
x
,
y
)
=
x
+
c
y
2
f_{XY}(x,y)=x+cy^2
f
X
Y
​
(
x
,
y
)
=
x
+
c
y
2
for
0
≤
x
≤
1
0\le x\le 1
0
≤
x
≤
1
and
0
≤
y
≤
1
0\le y\le 1
0
≤
y
≤
1
, and
0
0
0
otherwise. Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Numerical
EASY
3 marks
24 March 2024
45
Let
X
1
,
X
2
,
…
,
X
100
X_1, X_2, \ldots, X_{100}
X
1
​
,
X
2
​
,
…
,
X
100
​
be i.i.d. samples with mean
150
150
150
and variance
100
100
100
. Find the mean and variance of
X
ˉ
\bar{X}
X
ˉ
, where
X
ˉ
=
X
1
+
X
2
+
⋯
+
X
100
100
\bar{X}=\frac{X_1+X_2+\cdots+X_{100}}{100}
X
ˉ
=
100
X
1
​
+
X
2
​
+
⋯
+
X
100
​
​
is the sample mean.
Single correct
EASY
3 marks
24 March 2024
46
Find the marginal density of
X
X
X
.
Comprehension
MEDIUM
3 marks
24 March 2024
47
What is the value of
E
[
X
]
E[X]
E
[
X
]
? Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
24 March 2024
48
Let a random variable
X
X
X
denote the total number of students without laptops in the selected sample. Which of the following options is a good model for
X
X
X
?
Comprehension
EASY
2 marks
24 March 2024
49
Using the Central Limit Theorem, find an approximate probability that at least
80
80
80
students in the selected random sample own a laptop. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
3 marks
24 March 2024
50
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
24 March 2024
51
What is the value of
P
(
X
<
1
2
,
Y
<
1
2
)
P(X<\frac12, Y<\frac12)
P
(
X
<
2
1
​
,
Y
<
2
1
​
)
?
Comprehension
MEDIUM
3 marks
24 March 2024
52
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim \mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
, where
D
=
{
(
x
,
y
)
:
x
+
y
<
3
,
Â
x
>
0
,
Â
y
>
0
}
D=\{(x,y):x+y<3,\ x>0,\ y>0\}
D
=
{(
x
,
y
)
:
x
+
y
<
3
,
Â
x
>
0
,
Â
y
>
0
}
. What is the marginal density function of
f
X
(
x
)
f_X(x)
f
X
​
(
x
)
?
Single correct
MEDIUM
3 marks
03 December 2024
53
Suppose the moment generating function of a random variable
X
X
X
is given by
M
X
(
t
)
=
e
2
t
2
M_X(t)=e^{2t^2}
M
X
​
(
t
)
=
e
2
t
2
. Let
X
1
,
X
2
,
X
3
,
X
4
X_1,X_2,X_3,X_4
X
1
​
,
X
2
​
,
X
3
​
,
X
4
​
be i.i.d. samples with distribution
X
X
X
and
Y
=
X
1
+
X
2
+
X
3
+
X
4
Y=X_1+X_2+X_3+X_4
Y
=
X
1
​
+
X
2
​
+
X
3
​
+
X
4
​
. Which of the following options is true?
Single correct
EASY
3 marks
03 December 2024
54
Are
X
X
X
and
Y
Y
Y
independent?
Comprehension
EASY
2 marks
03 December 2024
55
Find the value of
P
(
X
<
Y
)
P(X<Y)
P
(
X
<
Y
)
. Enter the answer correct to one decimal place.
Comprehension
EASY
3 marks
03 December 2024
56
Using Central Limit theorem, find the approximate probability that there are more than
53
53
53
errors in a certain data file. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
03 December 2024
57
Find the joint density function
f
X
Y
(
x
,
y
)
f_{XY}(x,y)
f
X
Y
​
(
x
,
y
)
.
Comprehension
EASY
1 marks
06 August 2023
58
Find
P
(
∣
X
−
Y
∣
<
1
)
P(|X-Y|<1)
P
(
∣
X
−
Y
∣
<
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
4 marks
06 August 2023
59
Find the value of
c
c
c
. Enter the answer correct to two decimal places.
Comprehension
EASY
2 marks
06 August 2023
60
What is the probability that the pressure change during the chemical process is more than
0.5
0.5
0.5
, given that the temperature change is equal to
0.5
0.5
0.5
? Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
06 August 2023
61
Let a random variable
X
X
X
denote the total number of children who have normal eye-sight in the selected sample. Which of the following is true?
Comprehension
EASY
2 marks
06 August 2023
62
Using the Central Limit Theorem, find the approximate probability that in a random sample of
200
200
200
selected children at least
20
20
20
will have defective eye-sight. Enter the answer correct to 1 decimal place.
Comprehension
MEDIUM
3 marks
06 August 2023
63
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim \mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
, where
D
=
{
(
x
,
y
)
:
2
<
x
+
y
<
4
,
Â
x
>
0
,
Â
y
>
0
}
D=\{(x,y):2<x+y<4,\ x>0,\ y>0\}
D
=
{(
x
,
y
)
:
2
<
x
+
y
<
4
,
Â
x
>
0
,
Â
y
>
0
}
. Find
P
(
Y
<
2
X
)
P(Y<2X)
P
(
Y
<
2
X
)
.
Single correct
MEDIUM
3 marks
02 April 2023
64
Find the PDF of the marks of a candidate chosen uniformly at random.
Comprehension
MEDIUM
3 marks
02 April 2023
65
If a randomly selected candidate got
60
60
60
marks, what is the probability that the selected candidate is a boy?
Comprehension
MEDIUM
2 marks
02 April 2023
66
Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
2 marks
02 April 2023
67
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.
Comprehension
MEDIUM
2 marks
02 April 2023
68
Suppose a sample of
20
20
20
bottles is selected for a quality inspection. Let
X
X
X
denote the total number of bottles that are of bad quality in the selected sample. Which of the following is true?
Comprehension
EASY
2 marks
02 April 2023
69
Using the Central Limit Theorem, find the approximate probability that a machine will produce more than
370
370
370
bottles that are of good quality on a particular day. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
02 April 2023
70
Which of the following statements are true?
Comprehension
MEDIUM
3 marks
20 November 2022
71
Are
X
X
X
and
Y
Y
Y
independent?
Comprehension
EASY
1 marks
20 November 2022
72
Suppose iron rods are manufactured with a mean weight of
10
10
10
kg and a standard deviation of
0.4
0.4
0.4
kg. How does the variance of the sample mean change when the sample size is increased from
25
25
25
to
64
64
64
?
Single correct
EASY
3 marks
20 November 2022
73
Suppose
Y
Y
Y
represents the total number of bits without an error in a certain data file. Which of the following is true?
Comprehension
EASY
2 marks
20 November 2022
74
Using the Central Limit Theorem, find the approximate probability that there are more than
45
45
45
errors in a certain data file. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
20 November 2022
75
Find the value of
c
c
c
.
Comprehension
MEDIUM
2 marks
20 November 2022
76
Find the probability that the stock price of A will be higher than that of B. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
20 November 2022
77
Let
X
1
,
X
2
,
…
,
X
30
X_1,X_2,\ldots,X_{30}
X
1
​
,
X
2
​
,
…
,
X
30
​
be i.i.d. Poisson
(
2
)
(2)
(
2
)
, and let
Y
=
∑
i
=
1
30
X
i
Y=\sum_{i=1}^{30}X_i
Y
=
∑
i
=
1
30
​
X
i
​
. Using the Central Limit Theorem, find the value of
P
(
Y
>
20
)
P(Y>20)
P
(
Y
>
20
)
.
Single correct
MEDIUM
3 marks
10 July 2022
78
Suppose random variables
X
X
X
and
Y
Y
Y
are uniformly distributed over the region
D
=
(
[
−
1
,
0
]
×
[
0
,
1
]
)
∪
(
[
0
,
2
]
×
[
−
2
,
0
]
)
D=([-1,0]\times[0,1])\cup([0,2]\times[-2,0])
D
=
([
−
1
,
0
]
×
[
0
,
1
])
∪
([
0
,
2
]
×
[
−
2
,
0
])
. Choose the correct options from the following.
Multiple correct
MEDIUM
3 marks
10 July 2022
79
Calculate
P
(
0
<
X
<
1
/
2
,
Â
1
/
4
<
Y
<
1
/
2
)
P(0<X<1/2,\ 1/4<Y<1/2)
P
(
0
<
X
<
1/2
,
Â
1/4
<
Y
<
1/2
)
. Enter the answer correct to three decimal places.
Comprehension
EASY
1 marks
10 July 2022
80
Find
P
(
0
<
X
<
1
/
2
)
P(0<X<1/2)
P
(
0
<
X
<
1/2
)
. Enter the answer correct to one decimal place.
Comprehension
EASY
1 marks
10 July 2022
81
Find
P
(
X
<
2
Y
)
P(X<2Y)
P
(
X
<
2
Y
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
10 July 2022
82
Find the minimum value of
n
n
n
such that the probability that the sample mean
X
ˉ
\bar X
X
ˉ
is within
0.2
0.2
0.2
of the distribution mean is at least
0.9
0.9
0.9
using the Weak Law of Large Numbers.
Comprehension
MEDIUM
3 marks
10 July 2022
83
Find the distribution of runs of a randomly chosen player.
Comprehension
MEDIUM
3 marks
10 July 2022
Showing 83 questions.
Statistics 2 > Week 5
All PYQs
Topic-Wise PYQs
Start Weekly Test
Mock tests
Week mock
Topic mock
Subject mock
Type:
All
Difficulty:
All
Year:
All
83Q
01
The joint PDF of two continuous random variables
X
X
X
and
Y
Y
Y
is given by
f
X
Y
(
x
,
y
)
=
{
4
x
y
17
4
,
0
≤
x
≤
17
,
Â
0
≤
y
≤
17
,
0
,
otherwise.
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}
f
X
Y
​
(
x
,
y
)
=
⎩
⎨
⎧
​
1
7
4
4
x
y
​
,
0
,
​
0
≤
x
≤
17
,
Â
0
≤
y
≤
17
,
otherwise.
​
Are
X
X
X
and
Y
Y
Y
independent?
Single correct
MEDIUM
3 marks
06 April 2026
02
Let
X
1
,
X
2
,
X
3
,
X
4
X_1, X_2, X_3, X_4
X
1
​
,
X
2
​
,
X
3
​
,
X
4
​
be i.i.d. Normal random variables. Find the value of
P
(
3
X
1
+
X
2
+
2
X
3
+
3
X
4
≤
17
)
P(3X_1+X_2+2X_3+3X_4\le 17)
P
(
3
X
1
​
+
X
2
​
+
2
X
3
​
+
3
X
4
​
≤
17
)
.
Single correct
MEDIUM
3 marks
06 April 2026
03
Enter the correct option number for __A__.
Comprehension
EASY
0.5 marks
06 April 2026
04
Enter the correct option number for __B__.
Comprehension
EASY
0.5 marks
06 April 2026
05
Enter the correct option number for __C__.
Comprehension
EASY
0.5 marks
06 April 2026
06
Enter the correct option number for __D__.
Comprehension
EASY
0.5 marks
06 April 2026
07
Enter the correct option number for __E__.
Comprehension
EASY
0.5 marks
06 April 2026
08
Let
(
X
,
Y
)
∼
Uniform
(
R
)
(X,Y)\sim \text{Uniform}(R)
(
X
,
Y
)
∼
Uniform
(
R
)
, where
R
=
{
(
x
,
y
)
:
x
2
+
y
2
<
9
,
Â
x
>
0
,
Â
y
>
0
}
R=\{(x,y):x^2+y^2<9,\ x>0,\ y>0\}
R
=
{(
x
,
y
)
:
x
2
+
y
2
<
9
,
Â
x
>
0
,
Â
y
>
0
}
. Find the value of
P
(
X
<
1
,
Y
<
2
)
P(X<1,Y<2)
P
(
X
<
1
,
Y
<
2
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
2 marks
06 April 2026
09
Which of the following statements is/are correct?
Comprehension
MEDIUM
2 marks
06 April 2026
10
What is the marginal PDF of
X
X
X
at
x
=
50
x=50
x
=
50
?
Comprehension
MEDIUM
3 marks
06 April 2026
11
A randomly selected patient from this age group is found to have an enzyme level of
50
50
50
U/L. What is the probability that the patient is healthy?
Comprehension
MEDIUM
2 marks
06 April 2026
12
Which of the following statements is correct about the distribution of the sample mean?
Comprehension
EASY
2 marks
06 April 2026
13
Find the probability that the sample mean lifetime exceeds
506
506
506
hours. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
06 April 2026
14
Let
X
X
X
and
Y
Y
Y
be continuous random variables with joint PDF
f
X
,
Y
(
x
,
y
)
=
2
x
9
f_{X,Y}(x,y)=\frac{2x}{9}
f
X
,
Y
​
(
x
,
y
)
=
9
2
x
​
,
0
≤
x
≤
2
0\le x\le2
0
≤
x
≤
2
,
1
≤
y
≤
3
1\le y\le3
1
≤
y
≤
3
, and
0
0
0
otherwise. Find
P
(
X
<
1
∣
Y
=
2
)
P(X<1\mid Y=2)
P
(
X
<
1
∣
Y
=
2
)
correct to two decimal places.
Numerical
MEDIUM
3 marks
03 August 2025
15
What is the marginal density function
f
X
(
x
)
f_X(x)
f
X
​
(
x
)
?
Comprehension
MEDIUM
3 marks
03 August 2025
16
Find
P
(
0
<
X
<
1
,
Â
Y
<
1
)
P(0<X<1,\ Y<1)
P
(
0
<
X
<
1
,
Â
Y
<
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
03 August 2025
17
What is the marginal distribution of
Y
Y
Y
?
Comprehension
MEDIUM
2 marks
03 August 2025
18
Find
f
X
∣
Y
=
0.4
(
1
)
f_{X|Y=0.4}(1)
f
X
∣
Y
=
0.4
​
(
1
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
03 August 2025
19
Which option is an acceptable approximate model for
Y
Y
Y
as per the Central Limit Theorem?
Comprehension
MEDIUM
3 marks
03 August 2025
20
Using the Central Limit Theorem, it is calculated that
P
(
Y
>
a
)
≈
0.0228
P(Y>a)\approx0.0228
P
(
Y
>
a
)
≈
0.0228
. What is
a
a
a
? Use the useful standard normal data given in the question.
Comprehension
MEDIUM
2 marks
03 August 2025
21
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim\mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
where
D
=
{
(
x
,
y
)
:
x
+
y
<
2
,
x
>
0
,
y
>
0
}
D=\{(x,y):x+y<2,x>0,y>0\}
D
=
{(
x
,
y
)
:
x
+
y
<
2
,
x
>
0
,
y
>
0
}
. What is the marginal density function of
X
X
X
?
Single correct
MEDIUM
3 marks
16 March 2025
22
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
16 March 2025
23
What is
P
(
X
<
1
/
2
,
Y
<
1
/
2
)
P(X<1/2,Y<1/2)
P
(
X
<
1/2
,
Y
<
1/2
)
?
Comprehension
MEDIUM
3 marks
16 March 2025
24
Which option(s) is/are correct?
Comprehension
EASY
2 marks
16 March 2025
25
Which option is an acceptable approximate model for
S
S
S
as per the Central Limit Theorem?
Comprehension
EASY
2 marks
16 March 2025
26
Using CLT, find an approximate probability that
S
S
S
lies between
40
40
40
and
48
48
48
heads. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
16 March 2025
27
Let
X
1
,
X
2
,
…
,
X
10
X_1,X_2,\ldots,X_{10}
X
1
​
,
X
2
​
,
…
,
X
10
​
be i.i.d. random variables with common MGF
M
X
(
λ
)
M_X(\lambda)
M
X
​
(
λ
)
. Find the MGF of
X
ˉ
\bar X
X
ˉ
, where
X
ˉ
=
(
X
1
+
⋯
+
X
10
)
/
10
\bar X=(X_1+\cdots+X_{10})/10
X
ˉ
=
(
X
1
​
+
⋯
+
X
10
​
)
/10
.
Single correct
MEDIUM
3 marks
01 December 2024
28
Let
X
X
X
be study hours and
Y
Y
Y
recreation hours. The joint PDF is
f
X
Y
(
x
,
y
)
=
c
(
x
+
y
)
f_{XY}(x,y)=c(x+y)
f
X
Y
​
(
x
,
y
)
=
c
(
x
+
y
)
for
0
≤
x
≤
1
0\le x\le1
0
≤
x
≤
1
,
0
≤
y
≤
2
0\le y\le2
0
≤
y
≤
2
, and
0
0
0
otherwise. Find
21
c
21c
21
c
.
Numerical
EASY
3 marks
01 December 2024
29
Let
X
X
X
represent the total service time for
100
100
100
customers. Which option is an acceptable approximate model for
X
X
X
as per the CLT?
Comprehension
EASY
2 marks
01 December 2024
30
Using CLT, find the approximate probability that the total service time exceeds
1200
1200
1200
minutes. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
01 December 2024
31
Find the joint density function
f
X
Y
(
x
,
y
)
f_{XY}(x,y)
f
X
Y
​
(
x
,
y
)
.
Comprehension
EASY
3 marks
01 December 2024
32
Find
P
(
X
+
Y
<
2
)
P(X+Y<2)
P
(
X
+
Y
<
2
)
. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
3 marks
01 December 2024
33
A survey estimates the proportion of voters supporting a candidate. The true proportion is
0.45
0.45
0.45
and a random sample of
500
500
500
voters is taken. Using the Central Limit Theorem, what is the approximate probability that the sample proportion is between
0.43
0.43
0.43
and
0.47
0.47
0.47
?
Single correct
MEDIUM
3 marks
04 August 2024
34
If the joint PDF is
f
X
Y
(
x
,
y
)
=
1
/
k
f_{XY}(x,y)=1/k
f
X
Y
​
(
x
,
y
)
=
1/
k
for
(
x
,
y
)
∈
D
(x,y)\in D
(
x
,
y
)
∈
D
and
0
0
0
otherwise, find
k
k
k
correct to two decimal places.
Comprehension
EASY
1 marks
04 August 2024
35
Which option(s) are true?
Comprehension
MEDIUM
2 marks
04 August 2024
36
Find the conditional distribution
f
Y
∣
X
(
y
∣
x
)
f_{Y|X}(y|x)
f
Y
∣
X
​
(
y
∣
x
)
.
Comprehension
MEDIUM
3 marks
04 August 2024
37
Find
P
(
X
<
1
/
2
∣
Y
=
1
/
2
)
P(X<1/2\mid Y=1/2)
P
(
X
<
1/2
∣
Y
=
1/2
)
.
Comprehension
EASY
2 marks
04 August 2024
38
What is the expected value of the sample mean time spent on social media?
Comprehension
EASY
2 marks
04 August 2024
39
Using CLT, what is the approximate distribution of the sample mean recovery time?
Comprehension
EASY
2 marks
04 August 2024
40
Using CLT, find the approximate probability that the sample mean recovery time is between
4
4
4
and
6
6
6
days. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
04 August 2024
41
What is the distribution of
X
X
X
?
Comprehension
MEDIUM
3 marks
04 August 2024
42
Find
P
(
W
=
1
∣
X
=
−
1
)
P(W=1\mid X=-1)
P
(
W
=
1
∣
X
=
−
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
04 August 2024
43
Let
X
1
,
X
2
,
…
,
X
30
X_1, X_2, \ldots, X_{30}
X
1
​
,
X
2
​
,
…
,
X
30
​
be i.i.d. Bernoulli
(
0.8
)
(0.8)
(
0.8
)
. Define a new random variable
Y
=
X
1
+
X
2
+
⋯
+
X
30
Y=X_1+X_2+\cdots+X_{30}
Y
=
X
1
​
+
X
2
​
+
⋯
+
X
30
​
. Using the Central Limit Theorem, find the approximate value of
P
(
Y
>
21
)
P(Y>21)
P
(
Y
>
21
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
24 March 2024
44
Let
X
X
X
and
Y
Y
Y
be two continuous random variables with joint PDF
f
X
Y
(
x
,
y
)
=
x
+
c
y
2
f_{XY}(x,y)=x+cy^2
f
X
Y
​
(
x
,
y
)
=
x
+
c
y
2
for
0
≤
x
≤
1
0\le x\le 1
0
≤
x
≤
1
and
0
≤
y
≤
1
0\le y\le 1
0
≤
y
≤
1
, and
0
0
0
otherwise. Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Numerical
EASY
3 marks
24 March 2024
45
Let
X
1
,
X
2
,
…
,
X
100
X_1, X_2, \ldots, X_{100}
X
1
​
,
X
2
​
,
…
,
X
100
​
be i.i.d. samples with mean
150
150
150
and variance
100
100
100
. Find the mean and variance of
X
ˉ
\bar{X}
X
ˉ
, where
X
ˉ
=
X
1
+
X
2
+
⋯
+
X
100
100
\bar{X}=\frac{X_1+X_2+\cdots+X_{100}}{100}
X
ˉ
=
100
X
1
​
+
X
2
​
+
⋯
+
X
100
​
​
is the sample mean.
Single correct
EASY
3 marks
24 March 2024
46
Find the marginal density of
X
X
X
.
Comprehension
MEDIUM
3 marks
24 March 2024
47
What is the value of
E
[
X
]
E[X]
E
[
X
]
? Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
24 March 2024
48
Let a random variable
X
X
X
denote the total number of students without laptops in the selected sample. Which of the following options is a good model for
X
X
X
?
Comprehension
EASY
2 marks
24 March 2024
49
Using the Central Limit Theorem, find an approximate probability that at least
80
80
80
students in the selected random sample own a laptop. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
3 marks
24 March 2024
50
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
24 March 2024
51
What is the value of
P
(
X
<
1
2
,
Y
<
1
2
)
P(X<\frac12, Y<\frac12)
P
(
X
<
2
1
​
,
Y
<
2
1
​
)
?
Comprehension
MEDIUM
3 marks
24 March 2024
52
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim \mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
, where
D
=
{
(
x
,
y
)
:
x
+
y
<
3
,
Â
x
>
0
,
Â
y
>
0
}
D=\{(x,y):x+y<3,\ x>0,\ y>0\}
D
=
{(
x
,
y
)
:
x
+
y
<
3
,
Â
x
>
0
,
Â
y
>
0
}
. What is the marginal density function of
f
X
(
x
)
f_X(x)
f
X
​
(
x
)
?
Single correct
MEDIUM
3 marks
03 December 2024
53
Suppose the moment generating function of a random variable
X
X
X
is given by
M
X
(
t
)
=
e
2
t
2
M_X(t)=e^{2t^2}
M
X
​
(
t
)
=
e
2
t
2
. Let
X
1
,
X
2
,
X
3
,
X
4
X_1,X_2,X_3,X_4
X
1
​
,
X
2
​
,
X
3
​
,
X
4
​
be i.i.d. samples with distribution
X
X
X
and
Y
=
X
1
+
X
2
+
X
3
+
X
4
Y=X_1+X_2+X_3+X_4
Y
=
X
1
​
+
X
2
​
+
X
3
​
+
X
4
​
. Which of the following options is true?
Single correct
EASY
3 marks
03 December 2024
54
Are
X
X
X
and
Y
Y
Y
independent?
Comprehension
EASY
2 marks
03 December 2024
55
Find the value of
P
(
X
<
Y
)
P(X<Y)
P
(
X
<
Y
)
. Enter the answer correct to one decimal place.
Comprehension
EASY
3 marks
03 December 2024
56
Using Central Limit theorem, find the approximate probability that there are more than
53
53
53
errors in a certain data file. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
03 December 2024
57
Find the joint density function
f
X
Y
(
x
,
y
)
f_{XY}(x,y)
f
X
Y
​
(
x
,
y
)
.
Comprehension
EASY
1 marks
06 August 2023
58
Find
P
(
∣
X
−
Y
∣
<
1
)
P(|X-Y|<1)
P
(
∣
X
−
Y
∣
<
1
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
4 marks
06 August 2023
59
Find the value of
c
c
c
. Enter the answer correct to two decimal places.
Comprehension
EASY
2 marks
06 August 2023
60
What is the probability that the pressure change during the chemical process is more than
0.5
0.5
0.5
, given that the temperature change is equal to
0.5
0.5
0.5
? Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
06 August 2023
61
Let a random variable
X
X
X
denote the total number of children who have normal eye-sight in the selected sample. Which of the following is true?
Comprehension
EASY
2 marks
06 August 2023
62
Using the Central Limit Theorem, find the approximate probability that in a random sample of
200
200
200
selected children at least
20
20
20
will have defective eye-sight. Enter the answer correct to 1 decimal place.
Comprehension
MEDIUM
3 marks
06 August 2023
63
Let
(
X
,
Y
)
∼
U
n
i
f
o
r
m
(
D
)
(X,Y)\sim \mathrm{Uniform}(D)
(
X
,
Y
)
∼
Uniform
(
D
)
, where
D
=
{
(
x
,
y
)
:
2
<
x
+
y
<
4
,
Â
x
>
0
,
Â
y
>
0
}
D=\{(x,y):2<x+y<4,\ x>0,\ y>0\}
D
=
{(
x
,
y
)
:
2
<
x
+
y
<
4
,
Â
x
>
0
,
Â
y
>
0
}
. Find
P
(
Y
<
2
X
)
P(Y<2X)
P
(
Y
<
2
X
)
.
Single correct
MEDIUM
3 marks
02 April 2023
64
Find the PDF of the marks of a candidate chosen uniformly at random.
Comprehension
MEDIUM
3 marks
02 April 2023
65
If a randomly selected candidate got
60
60
60
marks, what is the probability that the selected candidate is a boy?
Comprehension
MEDIUM
2 marks
02 April 2023
66
Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
2 marks
02 April 2023
67
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.
Comprehension
MEDIUM
2 marks
02 April 2023
68
Suppose a sample of
20
20
20
bottles is selected for a quality inspection. Let
X
X
X
denote the total number of bottles that are of bad quality in the selected sample. Which of the following is true?
Comprehension
EASY
2 marks
02 April 2023
69
Using the Central Limit Theorem, find the approximate probability that a machine will produce more than
370
370
370
bottles that are of good quality on a particular day. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
02 April 2023
70
Which of the following statements are true?
Comprehension
MEDIUM
3 marks
20 November 2022
71
Are
X
X
X
and
Y
Y
Y
independent?
Comprehension
EASY
1 marks
20 November 2022
72
Suppose iron rods are manufactured with a mean weight of
10
10
10
kg and a standard deviation of
0.4
0.4
0.4
kg. How does the variance of the sample mean change when the sample size is increased from
25
25
25
to
64
64
64
?
Single correct
EASY
3 marks
20 November 2022
73
Suppose
Y
Y
Y
represents the total number of bits without an error in a certain data file. Which of the following is true?
Comprehension
EASY
2 marks
20 November 2022
74
Using the Central Limit Theorem, find the approximate probability that there are more than
45
45
45
errors in a certain data file. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
20 November 2022
75
Find the value of
c
c
c
.
Comprehension
MEDIUM
2 marks
20 November 2022
76
Find the probability that the stock price of A will be higher than that of B. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
20 November 2022
77
Let
X
1
,
X
2
,
…
,
X
30
X_1,X_2,\ldots,X_{30}
X
1
​
,
X
2
​
,
…
,
X
30
​
be i.i.d. Poisson
(
2
)
(2)
(
2
)
, and let
Y
=
∑
i
=
1
30
X
i
Y=\sum_{i=1}^{30}X_i
Y
=
∑
i
=
1
30
​
X
i
​
. Using the Central Limit Theorem, find the value of
P
(
Y
>
20
)
P(Y>20)
P
(
Y
>
20
)
.
Single correct
MEDIUM
3 marks
10 July 2022
78
Suppose random variables
X
X
X
and
Y
Y
Y
are uniformly distributed over the region
D
=
(
[
−
1
,
0
]
×
[
0
,
1
]
)
∪
(
[
0
,
2
]
×
[
−
2
,
0
]
)
D=([-1,0]\times[0,1])\cup([0,2]\times[-2,0])
D
=
([
−
1
,
0
]
×
[
0
,
1
])
∪
([
0
,
2
]
×
[
−
2
,
0
])
. Choose the correct options from the following.
Multiple correct
MEDIUM
3 marks
10 July 2022
79
Calculate
P
(
0
<
X
<
1
/
2
,
Â
1
/
4
<
Y
<
1
/
2
)
P(0<X<1/2,\ 1/4<Y<1/2)
P
(
0
<
X
<
1/2
,
Â
1/4
<
Y
<
1/2
)
. Enter the answer correct to three decimal places.
Comprehension
EASY
1 marks
10 July 2022
80
Find
P
(
0
<
X
<
1
/
2
)
P(0<X<1/2)
P
(
0
<
X
<
1/2
)
. Enter the answer correct to one decimal place.
Comprehension
EASY
1 marks
10 July 2022
81
Find
P
(
X
<
2
Y
)
P(X<2Y)
P
(
X
<
2
Y
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
10 July 2022
82
Find the minimum value of
n
n
n
such that the probability that the sample mean
X
ˉ
\bar X
X
ˉ
is within
0.2
0.2
0.2
of the distribution mean is at least
0.9
0.9
0.9
using the Weak Law of Large Numbers.
Comprehension
MEDIUM
3 marks
10 July 2022
83
Find the distribution of runs of a randomly chosen player.
Comprehension
MEDIUM
3 marks
10 July 2022
Showing 83 questions.
