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IITM BS Week 5 Statistics 2 Questions | Prasnya
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Type:
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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
,
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
,
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
Let
(
X
,
Y
)
∼
Uniform
(
R
)
(X,Y)\sim \text{Uniform}(R)
(
X
,
Y
)
∼
Uniform
(
R
)
, where
R
=
{
(
x
,
y
)
:
x
2
+
y
2
<
9
,
x
>
0
,
y
>
0
}
R=\{(x,y):x^2+y^2<9,\ x>0,\ y>0\}
R
=
{(
x
,
y
)
:
x
2
+
y
2
<
9
,
x
>
0
,
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
04
Enter the correct option number for __A__.
Comprehension
EASY
0.5 marks
06 April 2026
05
Enter the correct option number for __B__.
Comprehension
EASY
0.5 marks
06 April 2026
06
Enter the correct option number for __C__.
Comprehension
EASY
0.5 marks
06 April 2026
07
Enter the correct option number for __D__.
Comprehension
EASY
0.5 marks
06 April 2026
08
Enter the correct option number for __E__.
Comprehension
EASY
0.5 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
,
Y
<
1
)
P(0<X<1,\ Y<1)
P
(
0
<
X
<
1
,
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
,
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
,
x
>
0
,
y
>
0
}
D=\{(x,y):x+y<3,\ x>0,\ y>0\}
D
=
{(
x
,
y
)
:
x
+
y
<
3
,
x
>
0
,
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
28
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
29
Are
X
X
X
and
Y
Y
Y
independent?
Comprehension
EASY
2 marks
03 December 2024
30
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
31
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
32
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
33
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
34
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
35
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
36
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
37
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
38
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
39
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
40
Which option(s) are true?
Comprehension
MEDIUM
2 marks
04 August 2024
41
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
42
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
43
What is the expected value of the sample mean time spent on social media?
Comprehension
EASY
2 marks
04 August 2024
44
Using CLT, what is the approximate distribution of the sample mean recovery time?
Comprehension
EASY
2 marks
04 August 2024
45
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
46
What is the distribution of
X
X
X
?
Comprehension
MEDIUM
3 marks
04 August 2024
47
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
48
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
49
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
50
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
51
Find the marginal density of
X
X
X
.
Comprehension
MEDIUM
3 marks
24 March 2024
52
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
53
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
54
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
55
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
24 March 2024
56
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
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
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