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let y be a continuous random variable such that p
Question:
Let
Y
Y
Y
be a continuous random variable such that
P
(
Y
>
0
)
=
1
P(Y > 0) = 1
P
(
Y
>
0
)
=
1
and
E
(
Y
)
=
1
\mathbb{E}(Y) = 1
E
(
Y
)
=
1
. For
p
β
(
0
,
1
)
p \in (0, 1)
p
β
(
0
,
1
)
, let
ΞΎ
p
\xi_p
ΞΎ
p
β
denote the
p
p
p
th quantile of the probability distribution of the random variable
Y
Y
Y
. Then which of the following statements is always correct?
IIT JAM MS - 2024
IIT JAM MS
Updated On:
Oct 1, 2024
ΞΎ
0.75
β₯
5
\xi_{0.75} \geq 5
ΞΎ
0.75
β
β₯
5
ΞΎ
0.75
β€
4
\xi_{0.75} \leq 4
ΞΎ
0.75
β
β€
4
ΞΎ
0.25
β₯
4
\xi_{0.25} \geq 4
ΞΎ
0.25
β
β₯
4
ΞΎ
0.25
=
2
\xi_{0.25} = 2
ΞΎ
0.25
β
=
2
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The Correct Option is
B
Solution and Explanation
The correct option is (B):
ΞΎ
0.75
β€
4
\xi_{0.75} \leq 4
ΞΎ
0.75
β
β€
4
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Let
f
(
x
,
y
)
=
β£
x
y
β£
+
x
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(
x
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(
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β
R
2
(x, y) \in \mathbb{R}^2
(
x
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)
β
R
2
. Determine the existence of the partial derivative of
f
f
f
with respect to
x
x
x
exists:
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Let
f
(
x
)
=
4
x
2
β
sin
β‘
x
+
cos
β‘
2
x
f(x) = 4x^2 - \sin x + \cos 2x
f
(
x
)
=
4
x
2
β
sin
x
+
cos
2
x
for all
x
β
R
x \in \mathbb{R}
x
β
R
. Determine the properties of the function
f
f
f
:
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For
n
β
N
n \in \mathbb{N}
n
β
N
, let
a
n
=
n
sin
β‘
2
(
1
n
)
cos
β‘
n
,
a_n = \sqrt{n} \sin^2\left(\frac{1}{n}\right) \cos n,
a
n
β
=
n
β
sin
2
(
n
1
β
)
cos
n
,
and
b
n
=
n
sin
β‘
(
1
n
2
)
cos
β‘
n
.
b_n = \sqrt{n} \sin\left(\frac{1}{n^2}\right) \cos n.
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n
β
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2
1
β
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n
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Then
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{
(
x
1
,
x
2
,
x
3
,
x
4
)
β
R
4
β£
x
1
=
x
2
}
V = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1 = x_2 \}
V
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{(
x
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β
,
x
2
β
,
x
3
β
,
x
4
β
)
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R
4
β£
x
1
β
=
x
2
β
}
. Consider
V
V
V
as a subspace of
R
4
\mathbb{R}^4
R
4
over the real field. Then the dimension of
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V
V
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,
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Ξ©
=
{
1
,
2
,
3
,
4
,
5
,
6
}
. Then which of the following classes of sets is an algebra?
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