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01 — Article
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Quiz
Quiz: σ-Algebras and Probability Spaces
Module 1 of 5 · Easy
Quick Quiz
1.
Let
Ω
=
{
a
,
b
,
c
}
\Omega=\{a,b,c\}
Ω
=
{
a
,
b
,
c
}
. Which of the following collections is a valid σ-algebra on
Ω
\Omega
Ω
?
{
∅
,
{
a
}
,
{
b
}
,
{
c
}
,
Ω
}
\{\emptyset, \{a\}, \{b\}, \{c\}, \Omega\}
{
∅
,
{
a
}
,
{
b
}
,
{
c
}
,
Ω
}
{
∅
,
{
a
,
b
}
,
{
b
,
c
}
,
Ω
}
\{\emptyset, \{a,b\}, \{b,c\}, \Omega\}
{
∅
,
{
a
,
b
}
,
{
b
,
c
}
,
Ω
}
{
∅
,
Ω
}
\{\emptyset, \Omega\}
{
∅
,
Ω
}
{
∅
,
{
a
}
,
{
b
,
c
}
}
\{\emptyset, \{a\}, \{b,c\}\}
{
∅
,
{
a
}
,
{
b
,
c
}}
2.
The Borel σ-algebra
B
(
R
)
\mathcal{B}(\mathbb{R})
B
(
R
)
is generated by the open sets of
R
\mathbb{R}
R
. Which of the following generating families is equivalent?
Only
{
(
a
,
b
)
:
a
<
b
,
a
,
b
∈
Q
}
\{(a,b): a<b,\ a,b\in\mathbb{Q}\}
{(
a
,
b
)
:
a
<
b
,
a
,
b
∈
Q
}
(rational open intervals).
Only
{
(
−
∞
,
x
]
:
x
∈
R
}
\{(-\infty, x]: x\in\mathbb{R}\}
{(
−
∞
,
x
]
:
x
∈
R
}
(half-lines).
All of the above generate the same
B
(
R
)
\mathcal{B}(\mathbb{R})
B
(
R
)
.
Only
{
[
a
,
b
]
:
a
≤
b
}
\{[a,b]: a\le b\}
{[
a
,
b
]
:
a
≤
b
}
(closed intervals).
3.
Let
Ω
=
{
1
,
2
,
3
,
4
}
\Omega=\{1,2,3,4\}
Ω
=
{
1
,
2
,
3
,
4
}
and
X
:
Ω
→
R
X:\Omega\to\mathbb{R}
X
:
Ω
→
R
with
X
(
1
)
=
X
(
2
)
=
0
X(1)=X(2)=0
X
(
1
)
=
X
(
2
)
=
0
and
X
(
3
)
=
X
(
4
)
=
1
X(3)=X(4)=1
X
(
3
)
=
X
(
4
)
=
1
. What is
σ
(
X
)
\sigma(X)
σ
(
X
)
?
{
∅
,
Ω
}
\{\emptyset, \Omega\}
{
∅
,
Ω
}
{
∅
,
{
1
}
,
{
2
}
,
{
3
}
,
{
4
}
,
Ω
}
\{\emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \Omega\}
{
∅
,
{
1
}
,
{
2
}
,
{
3
}
,
{
4
}
,
Ω
}
{
∅
,
{
1
,
2
}
,
{
3
,
4
}
,
Ω
}
\{\emptyset, \{1,2\}, \{3,4\}, \Omega\}
{
∅
,
{
1
,
2
}
,
{
3
,
4
}
,
Ω
}
The full power set
2
Ω
2^\Omega
2
Ω
.
4.
A collection is closed under finite unions but not necessarily countable unions (all other axioms hold) — an algebra. What extra requirement upgrades an algebra to a σ-algebra?
Closure under uncountable unions.
Ω
\Omega
Ω
must be a countable set.
Closure under countable unions.
Every singleton
{
ω
}
\{\omega\}
{
ω
}
must belong to
F
\mathcal{F}
F
.
5.
Under
P
\mathbb{P}
P
on
Ω
=
{
H
,
T
}
\Omega=\{H,T\}
Ω
=
{
H
,
T
}
with
P
(
{
H
}
)
=
0.6
\mathbb{P}(\{H\})=0.6
P
({
H
})
=
0.6
,
P
(
{
T
}
)
=
0.4
\mathbb{P}(\{T\})=0.4
P
({
T
})
=
0.4
, the random variable
X
X
X
maps
H
↦
2
H\mapsto2
H
↦
2
,
T
↦
−
1
T\mapsto-1
T
↦
−
1
. What is
E
P
[
X
]
\mathbb{E}_\mathbb{P}[X]
E
P
[
X
]
?
$1.0$
$1.2$
$0.8$
$0.5$
6.
Why is it impossible to define a translation-invariant probability measure on ALL subsets of
[
0
,
1
]
[0,1]
[
0
,
1
]
?
Because the power set of
[
0
,
1
]
[0,1]
[
0
,
1
]
is not closed under countable unions.
Because every subset of
[
0
,
1
]
[0,1]
[
0
,
1
]
automatically has Lebesgue measure zero, forcing the total to vanish.
Because
[
0
,
1
]
[0,1]
[
0
,
1
]
is uncountable and measure must sum to 1 over atoms.
Because the Vitali set construction yields a non-measurable set.
7.
Which statement about completeness of a probability space is correct?
The measure assigns $0$ or $1$ to every event.
A complete probability space is one where
Ω
\Omega
Ω
is a complete metric space.
Every subset of a
P
\mathbb{P}
P
-null set belongs to
F
\mathcal{F}
F
.
The σ-algebra contains every open and closed subset of
R
\mathbb{R}
R
.
8.
A payoff depends on the closing price
S
T
S_T
S
T
and on whether the stock ever touched barrier
H
H
H
during
[
0
,
T
]
[0,T]
[
0
,
T
]
. It must be measurable with respect to which σ-algebra?
σ
(
S
t
,
0
≤
t
≤
T
)
\sigma(S_t,\ 0\le t\le T)
σ
(
S
t
,
0
≤
t
≤
T
)
, generated by the full price path.
The trivial σ-algebra
{
∅
,
Ω
}
\{\emptyset,\Omega\}
{
∅
,
Ω
}
.
σ
(
S
T
)
\sigma(S_T)
σ
(
S
T
)
, generated by the terminal price alone.
B
(
R
)
\mathcal{B}(\mathbb{R})
B
(
R
)
, the Borel σ-algebra on
R
\mathbb{R}
R
.
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