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Sub-Exponential Random Variables
Sub-Exponential Random Variables
1 questions
Difficulty 5-5
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Intermediate
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conceptual
The centered exponential distribution
Y
=
X
−
1
where
X
∼
Exp
(
1
)
is sub-exponential but NOT sub-Gaussian. What is the definitive evidence that
Y
cannot be sub-Gaussian?
Hide and think first
A.
The MGF
E
[
e
λY
]
diverges at
λ
=
1
, so no Gaussian-type MGF bound can hold
B.
The upper tail
P
(
Y
≥
t
)
decays only as
e
−
t
rather than
e
−
c
t
2
C.
The distribution is not symmetric, and a sub-Gaussian random variable must be symmetric by definition
D.
Its variance equals one, which is too large to satisfy a sub-Gaussian tail condition
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