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Statistics 2 > Week 3
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
37Q
01
Let
X
X
X
and
Y
Y
Y
be two independent random variables with
E
[
X
]
=
10
E[X]=10
E
[
X
]
=
10
,
E
[
Y
]
=
15
E[Y]=15
E
[
Y
]
=
15
,
Var
(
X
)
=
4
\operatorname{Var}(X)=4
Var
(
X
)
=
4
and
Var
(
Y
)
=
9
\operatorname{Var}(Y)=9
Var
(
Y
)
=
9
. Find the value of
E
[
(
X
+
Y
)
2
]
E[(X+Y)^2]
E
[(
X
+
Y
)
2
]
.
Numerical
MEDIUM
3 marks
26 October 2025
02
The probability mass function of a discrete random variable
X
X
X
is given by the table. Find the value of
b
b
b
such that
E
(
X
)
=
2.5
E(X)=2.5
E
(
X
)
=
2.5
. Enter the answer correct to one decimal place.
Numerical
MEDIUM
3 marks
26 October 2025
03
Using Markov's inequality, find an upper bound on the probability that the processing time is at least
1.3
1.3
1.3
times its expected value.
Comprehension
MEDIUM
2 marks
26 October 2025
04
Using Chebyshev's inequality, find a lower bound on the probability that the processing time lies between
480
480
480
milliseconds and
520
520
520
milliseconds. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
26 October 2025
05
Let
X
X
X
be the number of items produced by a machine on a given day with
E
[
X
]
=
7
E[X]=7
E
[
X
]
=
7
and
E
[
X
2
]
=
55
E[X^2]=55
E
[
X
2
]
=
55
. Using Chebyshev's inequality, find a lower bound on the probability that a machine produces between
4
4
4
and
10
10
10
items on a given day.
Single correct
MEDIUM
3 marks
13 July 2025
06
What is the expected payment that the designer earns for a randomly chosen project?
Comprehension
EASY
3 marks
13 July 2025
07
Using Markov's inequality, find an upper bound on the probability that the designer earns at least Rs.
15,000
15{,}000
15
,
000
. Enter the answer correct up to three decimal places.
Comprehension
MEDIUM
2 marks
13 July 2025
08
Calculate the value of
Var
(
2
X
+
1
)
\operatorname{Var}(2X+1)
Var
(
2
X
+
1
)
.
Comprehension
EASY
1 marks
13 July 2025
09
Find the correlation coefficient between
U
U
U
and
V
V
V
. Enter the answer correct up to two decimal places.
Comprehension
MEDIUM
4 marks
13 July 2025
10
A call center analyst monitors the number of calls received at the centre daily. Let
X
X
X
denote the number of calls received in a day. Using Markov's inequality, the analyst determines an upper bound of
0.48
0.48
0.48
on the probability of receiving more than
249
249
249
calls. Calculate the mean of
X
X
X
.
Numerical
MEDIUM
3 marks
23 February 2025
11
What is the value of
p
p
p
?
Comprehension
EASY
1 marks
23 February 2025
12
Find the expected amount of money to be gained. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
23 February 2025
13
Find
Var
(
W
)
\operatorname{Var}(W)
Var
(
W
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
23 February 2025
14
What is the value of
E
(
Z
+
W
)
E(Z+W)
E
(
Z
+
W
)
? Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
23 February 2025
15
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Find the value of
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
27 October 2024
16
Find the value of
E
[
T
]
E[T]
E
[
T
]
.
Comprehension
EASY
1 marks
27 October 2024
17
What is the value of
Var
(
3
T
+
4
)
\operatorname{Var}(3T+4)
Var
(
3
T
+
4
)
?
Comprehension
MEDIUM
2 marks
27 October 2024
18
Using Markov's inequality, find an upper bound on the probability that production will exceed
99
99
99
on a given day. Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
27 October 2024
19
Using Chebyshev's inequality, find a lower bound on the probability that on a given day production will be between
30
30
30
and
70
70
70
.
Comprehension
MEDIUM
3 marks
27 October 2024
20
Consider
n
n
n
bits
X
1
,
…
,
X
n
X_1,\ldots,X_n
X
1
,
…
,
X
n
, where each bit is equally likely to be
0
0
0
or
1
1
1
, and is independent of all other bits. Define
n
−
1
n-1
n
−
1
bits
Y
i
=
X
i
X
i
+
1
Y_i=X_iX_{i+1}
Y
i
=
X
i
X
i
+
1
,
i
=
1
,
…
,
n
−
1
i=1,\ldots,n-1
i
=
1
,
…
,
n
−
1
. Let
N
X
N_X
N
X
and
N
Y
N_Y
N
Y
be, respectively, the number of
1
1
1
s in
{
X
1
,
…
,
X
n
}
\{X_1,\ldots,X_n\}
{
X
1
,
…
,
X
n
}
and
{
Y
1
,
…
,
Y
n
−
1
}
\{Y_1,\ldots,Y_{n-1}\}
{
Y
1
,
…
,
Y
n
−
1
}
. Assuming
n
=
100
n=100
n
=
100
, what is the expected value of
N
X
N_X
N
X
and
N
Y
N_Y
N
Y
?
Single correct
MEDIUM
3 marks
07 July 2024
21
Using Markov inequality, find an upper bound to the probability
P
(
X
≥
35
)
P(X\ge35)
P
(
X
≥
35
)
. Which values below are greater than or equal to that upper bound?
Comprehension
MEDIUM
3 marks
07 July 2024
22
Using Chebyshev's inequality, find an upper bound to the probability that at least
35
35
35
people attended the interview. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
2 marks
07 July 2024
23
Find the expected value of
X
X
X
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
07 July 2024
24
Suppose
X
X
X
and
Y
Y
Y
are independent random variables with means
10
10
10
and
20
20
20
, and variances
2
2
2
and
4
4
4
, respectively. Find the value of
Var
(
X
Y
)
\operatorname{Var}(XY)
Var
(
X
Y
)
. Hint: If
X
X
X
and
Y
Y
Y
are independent, then
X
2
X^2
X
2
and
Y
2
Y^2
Y
2
are also independent.
Single correct
MEDIUM
3 marks
25 February 2024
25
The probability distribution of a discrete random variable
X
X
X
is given below. Calculate the expected value of
X
X
X
. Enter the answer correct to one decimal place.
Numerical
EASY
3 marks
25 February 2024
26
Using Markov's inequality, find a bound on the probability that this week's production will exceed
74
74
74
.
Comprehension
EASY
2 marks
25 February 2024
27
Using Chebyshev's inequality, find a lower bound on the probability that this week's production will be between
40
40
40
and
60
60
60
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
25 February 2024
28
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Find the value of
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
.
Numerical
MEDIUM
3 marks
29 October 2023
29
Using Chebyshev's inequality, find a lower bound for
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
P(-2\sigma\le X-\mu\le2\sigma)
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
, where
μ
\mu
μ
and
σ
2
\sigma^2
σ
2
are mean and variance of
X
X
X
, respectively. Enter the answer correct to two decimal places.
Numerical
MEDIUM
2 marks
29 October 2023
30
Find the expected value of
X
X
X
. Enter the answer correct to one decimal place.
Comprehension
EASY
1 marks
29 October 2023
31
The probability mass function of a random variable
X
X
X
is given as. Define
Y
=
(
2
X
+
1
)
2
Y=(2X+1)^2
Y
=
(
2
X
+
1
)
2
. Find the expected value of
Y
Y
Y
.
Numerical
EASY
3 marks
16 July 2023
32
Find the expected number of girls selected for the committee. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
16 July 2023
33
Using Chebyshev’s inequality, find a lower bound for
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
P(-2\sigma\le X-\mu\le 2\sigma)
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
, where
μ
\mu
μ
and
σ
2
\sigma^2
σ
2
are mean and variance of
X
X
X
. Enter the answer correct to
2
2
2
decimal places.
Comprehension
EASY
2 marks
16 July 2023
34
Let
X
X
X
be a Poisson random variable with mean equal to
20
20
20
. Which of the following bounds can be obtained using Markov’s inequality?
Single correct
MEDIUM
3 marks
16 October 2022
35
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Calculate
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
16 October 2022
36
Use Markov's inequality to find an upper bound on
P
(
X
≥
2040
)
P(X\ge2040)
P
(
X
≥
2040
)
. Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
37
Use Chebyshev's inequality to find a lower bound on
P
(
1988
<
X
<
2012
)
P(1988<X<2012)
P
(
1988
<
X
<
2012
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
Showing 37 questions.
Statistics 2 > Week 3
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
37Q
01
Let
X
X
X
and
Y
Y
Y
be two independent random variables with
E
[
X
]
=
10
E[X]=10
E
[
X
]
=
10
,
E
[
Y
]
=
15
E[Y]=15
E
[
Y
]
=
15
,
Var
(
X
)
=
4
\operatorname{Var}(X)=4
Var
(
X
)
=
4
and
Var
(
Y
)
=
9
\operatorname{Var}(Y)=9
Var
(
Y
)
=
9
. Find the value of
E
[
(
X
+
Y
)
2
]
E[(X+Y)^2]
E
[(
X
+
Y
)
2
]
.
Numerical
MEDIUM
3 marks
26 October 2025
02
The probability mass function of a discrete random variable
X
X
X
is given by the table. Find the value of
b
b
b
such that
E
(
X
)
=
2.5
E(X)=2.5
E
(
X
)
=
2.5
. Enter the answer correct to one decimal place.
Numerical
MEDIUM
3 marks
26 October 2025
03
Using Markov's inequality, find an upper bound on the probability that the processing time is at least
1.3
1.3
1.3
times its expected value.
Comprehension
MEDIUM
2 marks
26 October 2025
04
Using Chebyshev's inequality, find a lower bound on the probability that the processing time lies between
480
480
480
milliseconds and
520
520
520
milliseconds. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
26 October 2025
05
Let
X
X
X
be the number of items produced by a machine on a given day with
E
[
X
]
=
7
E[X]=7
E
[
X
]
=
7
and
E
[
X
2
]
=
55
E[X^2]=55
E
[
X
2
]
=
55
. Using Chebyshev's inequality, find a lower bound on the probability that a machine produces between
4
4
4
and
10
10
10
items on a given day.
Single correct
MEDIUM
3 marks
13 July 2025
06
What is the expected payment that the designer earns for a randomly chosen project?
Comprehension
EASY
3 marks
13 July 2025
07
Using Markov's inequality, find an upper bound on the probability that the designer earns at least Rs.
15,000
15{,}000
15
,
000
. Enter the answer correct up to three decimal places.
Comprehension
MEDIUM
2 marks
13 July 2025
08
Calculate the value of
Var
(
2
X
+
1
)
\operatorname{Var}(2X+1)
Var
(
2
X
+
1
)
.
Comprehension
EASY
1 marks
13 July 2025
09
Find the correlation coefficient between
U
U
U
and
V
V
V
. Enter the answer correct up to two decimal places.
Comprehension
MEDIUM
4 marks
13 July 2025
10
A call center analyst monitors the number of calls received at the centre daily. Let
X
X
X
denote the number of calls received in a day. Using Markov's inequality, the analyst determines an upper bound of
0.48
0.48
0.48
on the probability of receiving more than
249
249
249
calls. Calculate the mean of
X
X
X
.
Numerical
MEDIUM
3 marks
23 February 2025
11
What is the value of
p
p
p
?
Comprehension
EASY
1 marks
23 February 2025
12
Find the expected amount of money to be gained. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
23 February 2025
13
Find
Var
(
W
)
\operatorname{Var}(W)
Var
(
W
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
2 marks
23 February 2025
14
What is the value of
E
(
Z
+
W
)
E(Z+W)
E
(
Z
+
W
)
? Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
23 February 2025
15
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Find the value of
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
27 October 2024
16
Find the value of
E
[
T
]
E[T]
E
[
T
]
.
Comprehension
EASY
1 marks
27 October 2024
17
What is the value of
Var
(
3
T
+
4
)
\operatorname{Var}(3T+4)
Var
(
3
T
+
4
)
?
Comprehension
MEDIUM
2 marks
27 October 2024
18
Using Markov's inequality, find an upper bound on the probability that production will exceed
99
99
99
on a given day. Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
27 October 2024
19
Using Chebyshev's inequality, find a lower bound on the probability that on a given day production will be between
30
30
30
and
70
70
70
.
Comprehension
MEDIUM
3 marks
27 October 2024
20
Consider
n
n
n
bits
X
1
,
…
,
X
n
X_1,\ldots,X_n
X
1
,
…
,
X
n
, where each bit is equally likely to be
0
0
0
or
1
1
1
, and is independent of all other bits. Define
n
−
1
n-1
n
−
1
bits
Y
i
=
X
i
X
i
+
1
Y_i=X_iX_{i+1}
Y
i
=
X
i
X
i
+
1
,
i
=
1
,
…
,
n
−
1
i=1,\ldots,n-1
i
=
1
,
…
,
n
−
1
. Let
N
X
N_X
N
X
and
N
Y
N_Y
N
Y
be, respectively, the number of
1
1
1
s in
{
X
1
,
…
,
X
n
}
\{X_1,\ldots,X_n\}
{
X
1
,
…
,
X
n
}
and
{
Y
1
,
…
,
Y
n
−
1
}
\{Y_1,\ldots,Y_{n-1}\}
{
Y
1
,
…
,
Y
n
−
1
}
. Assuming
n
=
100
n=100
n
=
100
, what is the expected value of
N
X
N_X
N
X
and
N
Y
N_Y
N
Y
?
Single correct
MEDIUM
3 marks
07 July 2024
21
Using Markov inequality, find an upper bound to the probability
P
(
X
≥
35
)
P(X\ge35)
P
(
X
≥
35
)
. Which values below are greater than or equal to that upper bound?
Comprehension
MEDIUM
3 marks
07 July 2024
22
Using Chebyshev's inequality, find an upper bound to the probability that at least
35
35
35
people attended the interview. Enter the answer correct to one decimal place.
Comprehension
MEDIUM
2 marks
07 July 2024
23
Find the expected value of
X
X
X
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
07 July 2024
24
Suppose
X
X
X
and
Y
Y
Y
are independent random variables with means
10
10
10
and
20
20
20
, and variances
2
2
2
and
4
4
4
, respectively. Find the value of
Var
(
X
Y
)
\operatorname{Var}(XY)
Var
(
X
Y
)
. Hint: If
X
X
X
and
Y
Y
Y
are independent, then
X
2
X^2
X
2
and
Y
2
Y^2
Y
2
are also independent.
Single correct
MEDIUM
3 marks
25 February 2024
25
The probability distribution of a discrete random variable
X
X
X
is given below. Calculate the expected value of
X
X
X
. Enter the answer correct to one decimal place.
Numerical
EASY
3 marks
25 February 2024
26
Using Markov's inequality, find a bound on the probability that this week's production will exceed
74
74
74
.
Comprehension
EASY
2 marks
25 February 2024
27
Using Chebyshev's inequality, find a lower bound on the probability that this week's production will be between
40
40
40
and
60
60
60
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
25 February 2024
28
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Find the value of
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
.
Numerical
MEDIUM
3 marks
29 October 2023
29
Using Chebyshev's inequality, find a lower bound for
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
P(-2\sigma\le X-\mu\le2\sigma)
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
, where
μ
\mu
μ
and
σ
2
\sigma^2
σ
2
are mean and variance of
X
X
X
, respectively. Enter the answer correct to two decimal places.
Numerical
MEDIUM
2 marks
29 October 2023
30
Find the expected value of
X
X
X
. Enter the answer correct to one decimal place.
Comprehension
EASY
1 marks
29 October 2023
31
The probability mass function of a random variable
X
X
X
is given as. Define
Y
=
(
2
X
+
1
)
2
Y=(2X+1)^2
Y
=
(
2
X
+
1
)
2
. Find the expected value of
Y
Y
Y
.
Numerical
EASY
3 marks
16 July 2023
32
Find the expected number of girls selected for the committee. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
16 July 2023
33
Using Chebyshev’s inequality, find a lower bound for
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
P(-2\sigma\le X-\mu\le 2\sigma)
P
(
−
2
σ
≤
X
−
μ
≤
2
σ
)
, where
μ
\mu
μ
and
σ
2
\sigma^2
σ
2
are mean and variance of
X
X
X
. Enter the answer correct to
2
2
2
decimal places.
Comprehension
EASY
2 marks
16 July 2023
34
Let
X
X
X
be a Poisson random variable with mean equal to
20
20
20
. Which of the following bounds can be obtained using Markov’s inequality?
Single correct
MEDIUM
3 marks
16 October 2022
35
The joint PMF of two discrete random variables
X
X
X
and
Y
Y
Y
is given in the following table. Calculate
Cov
(
X
,
Y
)
\operatorname{Cov}(X,Y)
Cov
(
X
,
Y
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
16 October 2022
36
Use Markov's inequality to find an upper bound on
P
(
X
≥
2040
)
P(X\ge2040)
P
(
X
≥
2040
)
. Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
37
Use Chebyshev's inequality to find a lower bound on
P
(
1988
<
X
<
2012
)
P(1988<X<2012)
P
(
1988
<
X
<
2012
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
Showing 37 questions.
