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Concentration Inequalities
Concentration Inequalities
21 questions
Difficulty 2-10
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counterexample
A student claims: 'If two random variables
X
and
Y
are uncorrelated, then they are independent.' What is the simplest counterexample?
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A.
X
∼
Bernoulli
(
0.5
)
and
Y
=
1
−
X
: they have correlation
−
1
, which shows correlation captures all dependence
B.
X
∼
Uniform
{
−
1
,
0
,
1
}
and
Y
=
X
2
: covariance is zero because
E
[
X
Y
]
=
E
[
X
3
]
=
0
, but
Y
is determined by
X
C.
X
∼
N
(
0
,
1
)
and
Y
∼
N
(
0
,
1
)
independent: they are uncorrelated and independent, confirming the claim is correct
D.
X
and
Y
jointly uniform on the unit square
[
0
,
1
]
2
: they are uncorrelated because of symmetry and also clearly dependent
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