Bisher beschränkten wir uns auf ein abzählbares
Ω
{\displaystyle \Omega }
Ω
{\displaystyle \Omega }
1. Messung einer physikalischen Größe mit einer großen Genauigkeit. (
Ω
=
R
{\displaystyle \Omega =\mathbb {R} }
2. Exakter Zeitpunkt des Eintretens eines Erdbebenstoßes oder eines Telefonanrufs. (
Ω
=
R
+
{\displaystyle \Omega =\mathbb {R} _{+}}
3. Idealisiertes "stetiges" Roulette. (
Ω
=
[
0
,
2
π
[
{\displaystyle \Omega =[0,2\pi [}
4. Pseudo-Zufallszahlen. (
Ω
=
[
0
,
1
]
{\displaystyle \Omega =[0,1]}
In Beispiel 4. verlangen wir intuitiv von einer Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
[
0
,
1
]
{\displaystyle [0,1]}
P
(
[
a
,
b
]
)
=
b
−
a
,
0
≤
a
≤
b
≤
1
{\displaystyle P([a,b])=b-a,\quad 0\leq a\leq b\leq 1}
(*)
P
(
{
w
}
)
=
0
{\displaystyle P(\lbrace w\rbrace )=0}
Das mathematische Problem besteht nun darin, dass es keine Abbildung
P
:
S
(
[
0
,
1
]
)
→
[
0
,
1
]
{\displaystyle P:{\mathcal {S}}([0,1])\to [0,1]}
σ
{\displaystyle \sigma }
Statt auf ganz
S
(
Ω
)
{\displaystyle {\mathcal {S}}(\Omega )}
P
{\displaystyle P}
[
0
,
1
]
{\displaystyle [0,1]}
A
⊂
Ω
{\displaystyle A\subset \Omega }
S
⊂
S
(
Ω
)
{\displaystyle {\mathcal {S}}\subset {\mathcal {S}}(\Omega )}
S
{\displaystyle {\mathcal {S}}}
∩
,
∪
,
.
.
.
{\displaystyle \cap ,\cup ,...}
S
{\displaystyle {\mathcal {S}}}
Wenn man das Maß von Teilmengen
A
1
,
A
2
,
…
{\displaystyle A_{1},A_{2},\ldots }
Ω
{\displaystyle \Omega }
σ
{\displaystyle \sigma }
Ist
Ω
{\displaystyle \Omega }
S
⊂
S
(
Ω
)
{\displaystyle {\mathcal {S}}\subset {\mathcal {S}}(\Omega )}
σ
{\displaystyle \sigma }
Ω
{\displaystyle \Omega }
a)
Ω
∈
S
{\displaystyle \Omega \in {\mathcal {S}}}
b)
A
∈
S
⇒
A
¯
∈
S
{\displaystyle A\in {\mathcal {S}}\Rightarrow {\bar {A}}\in {\mathcal {S}}}
c)
A
1
,
A
2
,
.
.
.
∈
S
⇒
∪
i
=
1
∞
A
i
∈
U
{\displaystyle A_{1},A_{2},...\in {\mathcal {S}}\Rightarrow \cup _{i=1}^{\infty }A_{i}\in {\mathcal {U}}}
1. Es ist
∅
=
Ω
¯
∈
S
{\displaystyle \varnothing ={\bar {\Omega }}\in {\mathcal {S}}}
2.
A
1
,
A
2
,
.
.
.
∈
S
⇒
∩
i
=
1
∞
A
i
=
∪
i
=
1
∞
A
¯
i
∈
S
{\displaystyle A_{1},A_{2},...\in {\mathcal {S}}\Rightarrow \cap _{i=1}^{\infty }A_{i}=\cup _{i=1}^{\infty }{\bar {A}}_{i}\in {\mathcal {S}}}
3. In c) bzw 2. können wir auch
∪
i
=
1
∞
{\displaystyle \cup _{i=1}^{\infty }}
∩
i
=
1
∞
{\displaystyle \cap _{i=1}^{\infty }}
A
n
+
1
=
A
n
+
2
=
.
.
.
=
∅
{\displaystyle A_{n+1}=A_{n+2}=...=\varnothing }
A
n
+
1
=
A
n
+
2
=
.
.
.
=
Ω
{\displaystyle A_{n+1}=A_{n+2}=...=\Omega }
4.
B
(
Ω
)
{\displaystyle {\mathcal {B}}(\Omega )}
σ
{\displaystyle \sigma }
{
∅
,
Ω
}
{\displaystyle \lbrace \varnothing ,\Omega \rbrace }
σ
{\displaystyle \sigma }
5. Ist
I
⊂
S
(
Ω
)
{\displaystyle {\mathcal {I}}\subset {\mathcal {S}}(\Omega )}
σ
{\displaystyle \sigma }
I
{\displaystyle {\mathcal {I}}}
σ
{\displaystyle \sigma }
I
{\displaystyle {\mathcal {I}}}
σ
{\displaystyle \sigma }
σ
(
I
)
{\displaystyle \sigma ({\mathcal {I}})}
I
{\displaystyle {\mathcal {I}}}
σ
(
I
)
{\displaystyle \sigma ({\mathcal {I}})}
6. Ein Paar
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
S
{\displaystyle {\mathcal {S}}}
σ
{\displaystyle \sigma }
Ω
{\displaystyle \Omega }
Ein Triplet
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
a)
Ω
{\displaystyle \Omega }
b)
S
{\displaystyle {\mathcal {S}}}
σ
{\displaystyle \sigma }
Ω
{\displaystyle \Omega }
c)
P
:
S
→
[
0
,
1
]
{\displaystyle P:{\mathcal {S}}\to [0,1]}
(i)
P
(
Ω
)
=
1
{\displaystyle P(\Omega )=1}
(ii)
P
(
∪
i
=
1
∞
A
i
)
=
∑
i
=
1
∞
P
(
A
i
)
{\displaystyle P(\cup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i})}
A
1
,
A
2
,
.
.
.
∈
S
{\displaystyle A_{1},A_{2},...\in {\mathcal {S}}}
1.
P
{\displaystyle P}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
S
(
Ω
)
{\displaystyle {\mathcal {S}}(\Omega )}
σ
{\displaystyle \sigma }
S
{\displaystyle {\mathcal {S}}}
A
∈
S
{\displaystyle A\in {\mathcal {S}}}
A
∈
Ω
{\displaystyle A\in \Omega }
A
∈
S
(
Ω
)
{\displaystyle A\in {\mathcal {S}}(\Omega )}
2. Der diskrete Wahrscheinlichkeitsraum ergibt sich als Spezialfall der Definition:
Ω
{\displaystyle \Omega }
S
=
S
(
Ω
)
=
σ
{
{
w
}
w
∈
Ω
}
{\displaystyle {\mathcal {S}}={\mathcal {S}}(\Omega )=\sigma \lbrace \lbrace w\rbrace w\in \Omega \rbrace }
Konstruktion der Borelschen
σ
{\displaystyle \sigma }
Ω
=
R
k
,
k
≥
1
{\displaystyle \Omega =\mathbb {R} ^{k},k\geq 1}
I
k
⊂
S
(
R
k
)
{\displaystyle {\mathcal {I}}^{k}\subset {\mathcal {S}}(\mathbb {R} ^{k})}
a
=
(
a
1
,
.
.
.
a
k
)
,
b
=
(
b
1
,
.
.
.
,
b
k
)
∈
R
k
{\displaystyle a=(a_{1},...a_{k}),b=(b_{1},...,b_{k})\in \mathbb {R} ^{k}}
a
i
=
−
∞
{\displaystyle a_{i}=-\infty }
b
i
=
+
∞
{\displaystyle b_{i}=+\infty }
a
<
b
{\displaystyle a<b}
a
i
<
b
i
{\displaystyle a_{i}<b_{i}}
i
=
1
,
.
.
.
,
k
{\displaystyle i=1,...,k}
(
a
,
b
]
=
⊗
i
=
1
n
(
a
i
,
b
i
]
=
{
x
∈
R
k
:
a
i
<
x
i
<
b
i
{\displaystyle (a,b]=\otimes _{i=1}^{n}(a_{i},b_{i}]=\lbrace x\in \mathbb {R} ^{k}:a_{i}<x_{i}<b_{i}}
i
=
1
,
.
.
.
,
k
}
{\displaystyle i=1,...,k\rbrace }
I
k
=
{
(
a
,
b
]
,
a
<
b
}
{\displaystyle {\mathcal {I}}^{k}=\lbrace (a,b],a<b\rbrace }
R
k
∈
I
k
{\displaystyle \mathbb {R} ^{k}\in {\mathcal {I}}^{k}}
B
k
=
σ
(
I
k
)
{\displaystyle {\mathcal {B}}^{k}=\sigma ({\mathcal {I}}^{k})}
σ
{\displaystyle \sigma }
k
{\displaystyle k}
I
k
{\displaystyle {\mathcal {I}}^{k}}
B
{\displaystyle {\mathcal {B}}}
σ
{\displaystyle \sigma }
σ
{\displaystyle \sigma }
Satz aus der Topologie/Maßtheorie:
a) Die
σ
{\displaystyle \sigma }
B
k
{\displaystyle {\mathcal {B}}^{k}}
R
k
{\displaystyle \mathbb {R} ^{k}}
b) Es gibt nicht-Borelsche Mengen des
R
k
{\displaystyle \mathbb {R} ^{k}}
c)
B
k
{\displaystyle {\mathcal {B}}^{k}}
das System der offenen Intervalle
(
a
,
b
)
{\displaystyle (a,b)}
R
k
{\displaystyle \mathbb {R} ^{k}}
das System der abgeschlossenen Intervalle
[
a
,
b
]
{\displaystyle [a,b]}
R
k
{\displaystyle \mathbb {R} ^{k}}
das System der links abgeschlossenen und rechts offenen Mengen
[
a
,
b
)
{\displaystyle [a,b)}
R
k
{\displaystyle \mathbb {R} ^{k}}
Zur Festlegung einer Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
R
k
{\displaystyle \mathbb {R} ^{k}}
P
(
B
)
,
B
∈
B
k
{\displaystyle P(B),B\in {\mathcal {B}}^{k}}
Sei
P
~
:
I
k
→
[
0
,
1
]
{\displaystyle {\tilde {P}}:{\mathcal {I}}^{k}\to [0,1]}
i)
P
~
(
R
k
)
=
1
{\displaystyle {\tilde {P}}(\mathbb {R} ^{k})=1}
ii)
P
~
(
∪
i
=
1
∞
I
i
)
=
∑
i
=
1
∞
P
~
(
I
i
)
{\displaystyle {\tilde {P}}(\cup _{i=1}^{\infty }I_{i})=\sum _{i=1}^{\infty }{\tilde {P}}(I_{i})}
I
1
,
I
2
,
.
.
.
∈
I
k
{\displaystyle I_{1},I_{2},...\in {\mathcal {I}}^{k}}
∪
i
=
1
∞
I
i
∈
I
k
{\displaystyle \cup _{i=1}^{\infty }I_{i}\in {\mathcal {I}}^{k}}
Dann existiert genau eine Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
B
k
{\displaystyle {\mathcal {B}}^{k}}
P
|
I
k
=
P
~
{\displaystyle P|{\mathcal {I}}^{k}={\tilde {P}}}
P
(
I
)
=
P
~
(
I
)
{\displaystyle P(I)={\tilde {P}}(I)}
I
∈
I
k
{\displaystyle I\in {\mathcal {I}}^{k}}
P
{\displaystyle P}
P
~
{\displaystyle {\tilde {P}}}
B
k
{\displaystyle {\mathcal {B}}^{k}}
Öfter ist nun eine Teilmenge von
R
k
{\displaystyle \mathbb {R} ^{k}}
Ω
{\displaystyle \Omega }
Ω
=
[
0
,
1
]
k
{\displaystyle \Omega =[0,1]^{k}}
Ω
⊂
R
k
{\displaystyle \Omega \subset \mathbb {R} ^{k}}
Ω
∩
I
k
=
{
Ω
∩
I
,
I
∈
I
k
}
{\displaystyle \Omega \cap {\mathcal {I}}^{k}=\lbrace \Omega \cap I,I\in {\mathcal {I}}^{k}\rbrace }
I
k
{\displaystyle {\mathcal {I}}^{k}}
Ω
∩
B
k
=
{
Ω
∩
B
k
,
B
∈
B
k
}
{\displaystyle \Omega \cap {\mathcal {B}}^{k}=\lbrace \Omega \cap {\mathcal {B}}^{k},B\in {\mathcal {B}}^{k}\rbrace }
B
k
{\displaystyle {\mathcal {B}}^{k}}
Ω
{\displaystyle \Omega }
P
|
Ω
∩
B
k
{\displaystyle P|\Omega \cap {\mathcal {B}}^{k}}
P
{\displaystyle P}
P
{\displaystyle P}
Ω
∩
B
k
{\displaystyle \Omega \cap {\mathcal {B}}^{k}}
(
Ω
,
Ω
∩
B
k
,
P
|
Ω
∩
B
k
)
{\displaystyle (\Omega ,\Omega \cap {\mathcal {B}}^{k},P|\Omega \cap {\mathcal {B}}^{k})}
Ω
=
[
0
,
2
π
)
,
Ω
∩
I
1
=
{
(
a
,
b
]
,
0
≤
a
<
b
≤
2
π
}
{\displaystyle \Omega =[0,2\pi ),\Omega \cap {\mathcal {I}}^{1}=\lbrace (a,b],0\leq a<b\leq 2\pi \rbrace }
Durch
P
~
(
a
,
b
]
=
b
−
a
2
π
{\displaystyle {\tilde {P}}(a,b]={\frac {b-a}{2\pi }}}
Ω
∩
I
1
{\displaystyle \Omega \cap {\mathcal {I}}^{1}}
[
0
,
1
]
{\displaystyle [0,1]}
P
~
{\displaystyle {\tilde {P}}}
P
{\displaystyle P}
Ω
∩
B
1
{\displaystyle \Omega \cap {\mathcal {B}}^{1}}
[
0
,
2
π
)
{\displaystyle [0,2\pi )}
Zeitpunkt des Auftretens eines Ereignisses
Ω
=
[
0
,
∞
)
{\displaystyle \Omega =[0,\infty )}
P
~
(
a
,
b
]
=
e
−
λ
a
e
−
λ
b
,
0
≤
a
≤
b
≤
∞
{\displaystyle {\tilde {P}}(a,b]=e^{-\lambda a}e^{-\lambda b},0\leq a\leq b\leq \infty }
λ
>
0
{\displaystyle \lambda >0}
P
{\displaystyle P}
[
0
,
∞
)
∩
B
1
≡
B
+
1
{\displaystyle [0,\infty )\cap {\mathcal {B}}^{1}\equiv {\mathcal {B}}_{+}^{1}}
λ
{\displaystyle \lambda }
Zukünftig schreiben wir statt
P
~
{\displaystyle {\tilde {P}}}
P
{\displaystyle P}
Die Unabhängigkeit von Ereignissen
A
1
,
.
.
.
,
A
n
∈
S
{\displaystyle A_{1},...,A_{n}\in {\mathcal {S}}}
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
P
(
A
j
1
∩
.
.
.
∩
A
j
k
)
=
P
(
A
j
1
)
⋅
.
.
.
⋅
P
(
A
j
k
)
{\displaystyle P(A_{j_{1}}\cap ...\cap A_{j_{k}})=P(A_{j_{1}})\cdot ...\cdot P(A_{j_{k}})}
∅
≠
{
j
1
,
.
.
.
,
j
k
}
⊂
{
1
,
.
.
.
,
n
}
{\displaystyle \varnothing \neq \lbrace j_{1},...,j_{k}\rbrace \subset \lbrace 1,...,n\rbrace }
P
1
,
.
.
.
P
k
{\displaystyle P_{1},...P_{k}}
(
R
,
B
1
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}}^{1})}
P
{\displaystyle P}
(
R
k
,
B
j
k
)
{\displaystyle (\mathbb {R} ^{k},{\mathcal {B}}_{j}^{k})}
P
1
,
.
.
.
,
P
k
{\displaystyle P_{1},...,P_{k}}
P
=
P
1
×
.
.
.
×
P
k
{\displaystyle P=P_{1}\times ...\times P_{k}}
P
(
B
1
×
.
.
.
×
B
k
)
=
P
1
(
B
1
)
⋅
.
.
.
⋅
P
k
(
B
k
)
{\displaystyle P(B_{1}\times ...\times B_{k})=P_{1}(B_{1})\cdot ...\cdot P_{k}(B_{k})}
B
1
,
.
.
.
,
B
k
∈
B
1
{\displaystyle B_{1},...,B_{k}\in {\mathcal {B}}^{1}}
Der Begriff des Produktes von (allgemeinen) Wahrscheinlichkeitsräumen
(
Ω
i
,
S
i
,
P
i
)
{\displaystyle (\Omega _{i},{\mathcal {S}}_{i},P_{i})}
i
=
1
,
.
.
.
,
n
{\displaystyle i=1,...,n}
σ
{\displaystyle \sigma }
U
=
S
1
×
.
.
.
×
S
k
{\displaystyle {\mathcal {U}}={\mathcal {S}}_{1}\times ...\times {\mathcal {S}}_{k}}
S
i
=
B
1
,
U
=
B
k
{\displaystyle {\mathcal {S}}_{i}={\mathcal {B}}^{1},{\mathcal {U}}={\mathcal {B}}^{k}}
Bearbeiten
Der Begriff
P
(
B
|
A
)
{\displaystyle P(B|A)}
P
(
A
)
>
0
{\displaystyle P(A)>0}
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
A
,
B
,
A
1
,
.
.
.
,
A
m
{\displaystyle A,B,A_{1},...,A_{m}}
S
{\displaystyle {\mathcal {S}}}
Zunächst Beschränkung auf den Wahrscheinlichkeitsraum
(
R
,
B
1
,
P
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}}^{1},P)}
(
R
,
B
1
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}}^{1})}
(
R
∩
Ω
,
B
1
∩
Ω
)
{\displaystyle (\mathbb {R} \cap \Omega ,{\mathcal {B}}^{1}\cap \Omega )}
(
a
,
b
]
=
(
−
∞
,
b
]
∖
(
−
∞
,
a
]
{\displaystyle (a,b]=(-\infty ,b]\setminus (-\infty ,a]}
P
(
a
,
b
]
=
P
(
−
∞
,
b
]
−
P
(
−
∞
,
a
]
{\displaystyle P(a,b]=P(-\infty ,b]-P(-\infty ,a]}
F
(
t
)
=
P
(
−
∞
,
t
]
,
t
∈
R
{\displaystyle F(t)=P(-\infty ,t],t\in \mathbb {R} }
Bearbeiten
Sei
P
{\displaystyle P}
(
R
,
B
1
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}}^{1})}
F
P
:
R
→
[
0
,
1
]
,
F
P
(
t
)
=
P
(
−
∞
,
t
]
,
t
∈
R
{\displaystyle F_{P}:\mathbb {R} \to [0,1],F_{P}(t)=P(-\infty ,t],t\in \mathbb {R} }
P
{\displaystyle P}
(Im Folgenden sei
F
(
t
+
)
=
l
i
m
s
↓
t
F
(
s
)
,
F
(
t
−
)
=
l
i
m
s
↑
t
F
(
s
)
{\displaystyle F(t+)=lim_{s\downarrow t}F(s),F(t-)=lim_{s\uparrow t}F(s)}
Sei
F
(
t
)
≡
F
P
(
t
)
,
t
∈
R
{\displaystyle F(t)\equiv F_{P}(t),t\in \mathbb {R} }
P
{\displaystyle P}
i)
F
(
t
)
{\displaystyle F(t)}
0
≤
F
(
t
)
≤
1
{\displaystyle 0\leq F(t)\leq 1}
ii)
F
(
t
+
)
=
F
(
t
)
{\displaystyle F(t+)=F(t)}
iii)
F
(
t
−
)
=
F
(
t
)
−
P
(
{
w
}
)
{\displaystyle F(t-)=F(t)-P(\lbrace w\rbrace )}
iv)
l
i
m
t
→
−
∞
F
(
t
)
=
0
,
l
i
m
t
→
∞
F
(
t
)
=
1
{\displaystyle lim_{t\to -\infty }F(t)=0,lim_{t\to \infty }F(t)=1}
i) Monotonieeigenschaft von
P
{\displaystyle P}
ii) Sei
t
n
↓
t
(
t
n
>
t
)
{\displaystyle t_{n}\downarrow t(t_{n}>t)}
(
t
,
t
n
]
=
⋃
i
=
n
∞
(
t
i
+
1
,
t
i
]
{\displaystyle (t,t_{n}]=\bigcup _{i=n}^{\infty }(t_{i+1},t_{i}]}
F
(
t
+
)
=
l
i
m
n
→
∞
F
(
t
n
)
=
l
i
m
n
→
∞
[
F
(
t
)
+
P
(
t
,
t
n
]
]
{\displaystyle F(t+)=lim_{n\to \infty }F(t_{n})=lim_{n\to \infty }[F(t)+P(t,t_{n}]]}
=
l
i
m
n
→
∞
[
F
(
t
)
+
∑
i
=
n
∞
P
(
t
i
+
1
,
t
i
]
]
=
F
(
t
)
+
0
{\displaystyle =lim_{n\to \infty }[F(t)+\sum _{i=n}^{\infty }P(t_{i+1},t_{i}]]=F(t)+0}
da die Reihe
∑
i
=
1
∞
P
(
t
i
+
1
,
t
i
]
=
P
(
t
,
t
1
]
<
∞
{\displaystyle \sum _{i=1}^{\infty }P(t_{i+1},t_{i}]=P(t,t_{1}]<\infty }
iii) Sei
t
n
↑
t
(
t
n
<
t
)
{\displaystyle t_{n}\uparrow t(t_{n}<t)}
(
t
1
,
t
n
=
⋃
i
=
1
n
−
1
(
t
i
,
t
i
+
1
]
{\displaystyle (t_{1},t_{n}=\bigcup _{i=1}^{n-1}(t_{i},t_{i+1}]}
F
(
t
−
)
=
l
i
m
n
→
∞
F
(
t
n
)
=
l
i
m
n
→
∞
[
F
(
t
1
)
+
P
(
t
1
,
t
n
]
]
{\displaystyle F(t-)=lim_{n\to \infty }F(t_{n})=lim_{n\to \infty }[F(t_{1})+P(t_{1},t_{n}]]}
=
l
i
m
n
→
∞
[
F
(
t
1
1
)
+
∑
i
=
1
n
−
1
P
(
t
i
,
t
i
+
1
]
]
{\displaystyle =lim_{n\to \infty }[F(t_{1}1)+\sum _{i=1}^{n-1}P(t_{i},t_{i+1}]]}
=
F
(
t
1
)
+
P
(
⋃
i
=
1
∞
(
t
i
,
t
i
+
1
]
)
=
F
(
t
1
)
+
P
(
t
1
,
t
)
{\displaystyle =F(t_{1})+P(\bigcup _{i=1}^{\infty }(t_{i},t_{i+1}])=F(t_{1})+P(t_{1},t)}
=
P
(
−
∞
,
t
)
=
F
(
t
)
−
P
(
{
w
}
)
{\displaystyle =P(-\infty ,t)=F(t)-P(\lbrace w\rbrace )}
iv) Analog zu ii) und iii).
Die Limiten in ii), iii), iv) existieren wegen i).
Im Folgenden bezeichne
⟨
a
,
b
⟩
{\displaystyle \langle a,b\rangle }
−
∞
≤
a
<
b
≤
∞
{\displaystyle -\infty \leq a<b\leq \infty }
[
a
,
b
]
,
(
a
,
b
)
,
[
a
,
b
)
,
(
a
,
b
]
{\displaystyle [a,b],(a,b),[a,b),(a,b]}
a
=
−
∞
{\displaystyle a=-\infty }
(
−
∞
,
b
⟩
{\displaystyle (-\infty ,b\rangle }
b
=
∞
{\displaystyle b=\infty }
⟨
a
,
∞
)
{\displaystyle \langle a,\infty )}
Sei
F
{\displaystyle F}
P
{\displaystyle P}
P
(
a
,
b
]
=
F
(
b
)
−
F
(
a
)
{\displaystyle P(a,b]=F(b)-F(a)}
P
(
−
∞
,
b
)
=
F
(
b
)
{\displaystyle P(-\infty ,b)=F(b)}
P
(
a
,
b
)
=
F
(
b
−
)
−
F
(
a
)
{\displaystyle P(a,b)=F(b-)-F(a)}
P
(
a
,
∞
)
=
1
−
F
(
a
)
{\displaystyle P(a,\infty )=1-F(a)}
P
[
a
,
b
]
=
P
(
b
)
−
P
(
a
−
)
{\displaystyle P[a,b]=P(b)-P(a-)}
[
a
,
b
)
=
F
(
b
−
)
−
F
(
a
−
)
{\displaystyle [a,b)=F(b-)-F(a-)}
1. Falls
F
{\displaystyle F}
a
{\displaystyle a}
F
{\displaystyle F}
[
a
,
b
]
{\displaystyle [a,b]}
P
[
a
,
b
]
=
0
{\displaystyle P[a,b]=0}
2. Zusammen mit dem Fortsetzungssatz folgt, dass
P
{\displaystyle P}
F
(
t
)
,
t
∈
R
{\displaystyle F(t),t\in \mathbb {R} }
P
(
a
,
b
]
=
F
(
b
)
−
F
(
a
)
{\displaystyle P(a,b]=F(b)-F(a)}
Im Fall der Exponentialverteilung aus dem Beispiel 1.5.3, bei der
F
(
t
)
=
{
0
,
t
<
0
λ
e
−
λ
t
,
t
≥
0
{\displaystyle F(t)=\left\{{\begin{array}{ll}0,&t<0\\\lambda e^{-\lambda t},&t\geq 0\end{array}}\right.}
F
′
=
f
{\displaystyle F'=f}
F
(
t
)
=
∫
−
∞
t
f
(
x
)
d
x
{\displaystyle F(t)=\int _{-\infty }^{t}f(x)dx}
f
(
x
)
=
{
0
,
x
<
0
λ
e
−
λ
t
,
x
≥
0
{\displaystyle f(x)=\left\{{\begin{array}{ll}0,&x<0\\\lambda e^{-\lambda t},&x\geq 0\end{array}}\right.}
Sei
F
P
(
t
)
,
t
∈
R
{\displaystyle F_{P}(t),t\in \mathbb {R} }
P
{\displaystyle P}
f
P
:
R
→
[
0
,
∞
)
{\displaystyle f_{P}:\mathbb {R} \to [0,\infty )}
F
P
(
t
)
=
∫
−
∞
∞
f
P
(
x
)
d
x
{\displaystyle F_{P}(t)=\int _{-\infty }^{\infty }f_{P}(x)dx}
t
∈
R
{\displaystyle t\in \mathbb {R} }
f
P
(
x
)
,
x
∈
R
{\displaystyle f_{P}(x),x\in \mathbb {R} }
P
{\displaystyle P}
1. Das Integral
∫
−
∞
∞
{\displaystyle \int _{-\infty }^{\infty }}
2. Ist die stetige Verteilungsfunktion
F
{\displaystyle F}
R
∖
D
{\displaystyle \mathbb {R} \setminus D}
D
{\displaystyle D}
F
{\displaystyle F}
f
(
x
)
=
F
′
(
x
)
,
x
∈
R
∖
D
{\displaystyle f(x)=F'(x),x\in \mathbb {R} \setminus D}
f
(
x
)
{\displaystyle f(x)}
D
{\displaystyle D}
3. Besitzt
F
{\displaystyle F}
F
(
t
)
,
t
∈
R
{\displaystyle F(t),t\in \mathbb {R} }
F
(
t
)
=
F
(
t
+
)
=
F
(
t
−
)
{\displaystyle F(t)=F(t+)=F(t-)}
P
⟨
a
,
b
⟩
{\displaystyle P\langle a,b\rangle }
P
⟨
a
,
b
⟩
=
F
(
b
)
−
F
(
a
)
{\displaystyle P\langle a,b\rangle =F(b)-F(a)}
Besitzt eine Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
(
R
,
B
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}})}
f
=
f
P
{\displaystyle f=f_{P}}
P
⟨
a
,
b
⟩
=
∫
a
b
f
(
x
)
d
x
{\displaystyle P\langle a,b\rangle =\int _{a}^{b}f(x)dx}
Insbesondere gilt:
∫
−
∞
∞
f
(
x
)
d
x
=
1
{\displaystyle \int _{-\infty }^{\infty }f(x)dx=1}
Folgt direkt aus Bemerkung 3.
1. Wir können also eine Dichte
f
{\displaystyle f}
f
:
R
∈
[
0
,
∞
)
{\displaystyle f:\mathbb {R} \in [0,\infty )}
f
{\displaystyle f}
∫
−
∞
∞
f
(
x
)
d
x
=
1
{\displaystyle \int _{-\infty }^{\infty }f(x)dx=1}
2. Durch Vorgabe einer Dichte
f
{\displaystyle f}
P
{\displaystyle P}
(
R
,
B
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}})}
3. Der Begriff der Dichte spielt im Fall
Ω
=
R
{\displaystyle \Omega =\mathbb {R} }
Ω
{\displaystyle \Omega }
Gleichverteilung auf dem Intervall
Ω
=
[
A
,
B
]
⊂
R
,
A
<
B
{\displaystyle \Omega =[A,B]\subset \mathbb {R} ,A<B}
Dichte:
f
(
x
)
=
{
0
,
x
∉
[
A
,
B
]
1
B
−
A
,
x
∈
[
A
,
B
]
{\displaystyle f(x)=\left\{{\begin{array}{ll}0,&x\notin [A,B]\\{\frac {1}{B-A}},&x\in [A,B]\end{array}}\right.}
Verteilungsfunktion:
F
(
x
)
=
{
0
,
x
<
A
x
−
A
B
−
A
,
A
≤
x
≤
B
1
,
x
>
B
{\displaystyle F(x)=\left\{{\begin{array}{ll}0,&x<A\\{\frac {x-A}{B-A}},&A\leq x\leq B\\1,&x>B\end{array}}\right.}
Exponentialverteilung mit dem Parameter
λ
>
0
{\displaystyle \lambda >0}
Dichte:
f
(
x
)
=
{
0
,
x
<
0
λ
e
−
λ
x
,
x
≥
0
{\displaystyle f(x)=\left\{{\begin{array}{ll}0,&x<0\\\lambda e^{-\lambda x},&x\geq 0\end{array}}\right.}
Verteilungsfunktion:
F
(
x
)
=
{
0
,
x
<
0
1
−
e
−
λ
e
,
x
≥
0
{\displaystyle F(x)=\left\{{\begin{array}{ll}0,&x<0\\1-e^{-\lambda e},&x\geq 0\end{array}}\right.}
Verwendung:
Wartezeit (bis zum Eintreten eines Ereignisses). Diskrete Verteilung auf
{
x
1
,
x
2
,
.
.
.
}
⊂
R
{\displaystyle \lbrace x_{1},x_{2},...\rbrace \subset \mathbb {R} }
{
x
1
,
.
.
.
,
x
n
}
⊂
R
{\displaystyle \lbrace x_{1},...,x_{n}\rbrace \subset \mathbb {R} }
P
{
x
i
}
{\displaystyle P\lbrace x_{i}\rbrace }
A
∈
B
1
{\displaystyle A\in {\mathcal {B}}^{1}}
P
(
A
)
=
∑
i
,
x
i
≤
x
P
(
{
x
i
}
)
{\displaystyle P(A)=\sum _{i,x_{i}\leq x}P(\lbrace x_{i}\rbrace )}
P
{\displaystyle P}
(
R
,
B
)
{\displaystyle (\mathbb {R} ,{\mathcal {B}})}
F
(
x
)
=
∑
i
:
x
i
≤
x
P
(
{
x
i
}
)
{\displaystyle F(x)=\sum _{i:x_{i}\leq x}P(\lbrace x_{i}\rbrace )}
Es existiert jedoch keine Dichte!
Normalverteilung mit Parametern
μ
{\displaystyle \mu }
σ
2
,
μ
∈
R
,
σ
2
>
0
{\displaystyle \sigma ^{2},\mu \in \mathbb {R} ,\sigma ^{2}>0}
Dichte:
ϕ
μ
,
σ
2
(
x
)
=
1
2
π
σ
e
1
2
⋅
(
x
−
μ
σ
)
2
{\displaystyle \phi _{\mu ,\sigma ^{2}}(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{{\frac {1}{2}}\cdot ({\frac {x-\mu }{\sigma }})^{2}}}
Verteilungsfunktion:
Φ
μ
,
σ
2
(
x
)
=
∫
−
∞
x
ϕ
μ
,
σ
2
(
t
)
d
t
{\displaystyle \Phi _{\mu ,\sigma ^{2}}(x)=\int _{-\infty }^{x}\phi _{\mu ,\sigma ^{2}}(t)dt}
Abkürzung:
N
(
μ
,
σ
2
)
{\displaystyle N(\mu ,\sigma ^{2})}
Verwendung: Symmetrisch um einen 'wahren' Wert
μ
{\displaystyle \mu }
Spezialfall:
N
(
0
,
1
)
{\displaystyle N(0,1)}
ϕ
=
ϕ
0
,
1
,
Φ
=
Φ
0
,
1
{\displaystyle \phi =\phi _{0,1},\Phi =\Phi _{0,1}}
ϕ
μ
,
σ
2
(
x
)
=
1
σ
ϕ
(
x
−
μ
σ
)
,
Φ
μ
,
σ
2
(
x
)
=
Φ
(
x
−
μ
σ
)
{\displaystyle \phi _{\mu ,\sigma ^{2}}(x)={\frac {1}{\sigma }}\phi ({\frac {x-\mu }{\sigma }}),\Phi _{\mu ,\sigma ^{2}}(x)=\Phi ({\frac {x-\mu }{\sigma }})}
Aus dieser Beziehung folgt:
∫
−
∞
∞
ϕ
μ
,
σ
2
(
x
)
d
x
=
Φ
μ
,
σ
2
(
∞
)
=
Φ
(
∞
)
=
∫
−
∞
∞
ϕ
(
x
)
d
x
=
1
,
{\displaystyle \int _{-\infty }^{\infty }\phi _{\mu ,\sigma ^{2}}(x)dx=\Phi _{\mu ,\sigma ^{2}}(\infty )=\Phi (\infty )=\int _{-\infty }^{\infty }\phi (x)dx=1,}
so dass
ϕ
μ
,
σ
2
{\displaystyle \phi _{\mu ,\sigma ^{2}}}
Das Konzept der Dichte lässt sich auch im Fall
Ω
=
R
k
{\displaystyle \Omega =\mathbb {R} ^{k}}
R
k
{\displaystyle \mathbb {R} ^{k}}
f
(
x
)
=
f
(
x
1
,
.
.
.
,
x
k
)
,
x
∈
R
{\displaystyle f(x)=f(x_{1},...,x_{k}),x\in \mathbb {R} }
∫
R
k
f
(
x
1
,
.
.
.
,
x
k
)
d
k
x
=
1
{\displaystyle \int _{\mathbb {R} ^{k}}f(x_{1},...,x_{k})d^{k}x=1}
Für ein
B
∈
B
k
{\displaystyle B\in {\mathcal {B}}^{k}}
∫
B
f
(
x
1
,
.
.
.
,
x
k
)
d
k
x
=
∫
R
k
f
(
x
)
d
k
x
=
∫
R
k
1
B
(
x
)
f
(
x
)
d
k
x
.
{\displaystyle \int _{B}f(x_{1},...,x_{k})d^{k}x=\int _{\mathbb {R} ^{k}}f(x)d^{k}x=\int _{\mathbb {R} ^{k}}1_{B}(x)f(x)d^{k}x.}
Wir benötigen den folgenden Satz der Integrationstheorie.
Ist
f
≥
0
{\displaystyle f\geq 0}
R
k
{\displaystyle \mathbb {R} ^{k}}
B
↦
∫
B
f
(
x
)
d
k
x
,
b
∈
B
k
{\displaystyle B\mapsto \int _{B}f(x)d^{k}x,b\in {\mathcal {B}}^{k}}
σ
{\displaystyle \sigma }
B
k
{\displaystyle {\mathcal {B}}^{k}}
[
0
,
∞
)
{\displaystyle [0,\infty )}
B
1
,
.
.
.
,
B
k
∈
B
k
{\displaystyle B_{1},...,B_{k}\in {\mathcal {B}}^{k}}
∫
⋃
B
i
f
(
x
)
d
k
x
=
∑
i
=
1
∞
∫
B
i
f
(
x
)
d
k
x
{\displaystyle \int _{\bigcup B_{i}}f(x)d^{k}x=\sum _{i=1}^{\infty }\int _{B_{i}}f(x)d^{k}x}
Über den Satz der monotonen Konvergenz.
Sei
f
(
x
)
,
x
∈
R
k
{\displaystyle f(x),x\in \mathbb {R} ^{k}}
(
a
,
b
]
{\displaystyle (a,b]}
n
{\displaystyle n}
(
a
,
b
]
=
⊗
i
=
1
k
(
a
i
,
b
i
]
{\displaystyle (a,b]=\otimes _{i=1}^{k}(a_{i},b_{i}]}
a) Setzt man
(*)
P
(
a
,
b
]
=
∫
(
a
,
b
]
f
(
x
)
d
k
x
{\displaystyle P(a,b]=\int _{(a,b]}f(x)d^{k}x}
so wird eine Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
(
R
k
,
B
k
)
{\displaystyle (\mathbb {R} ^{k},{\mathcal {B}}^{k})}
(
a
,
b
]
{\displaystyle (a,b]}
⟨
a
,
b
⟩
=
⊗
i
=
1
k
⟨
a
i
,
b
i
⟩
{\displaystyle \langle a,b\rangle =\otimes _{i=1}^{k}\langle a_{i},b_{i}\rangle }
b) Für die Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
P
(
B
)
=
∫
B
f
(
x
)
d
k
x
(
B
∈
B
k
)
{\displaystyle P(B)=\int _{B}f(x)d^{k}x(B\in {\mathcal {B}}^{k})}
a) Durch (*) wird eine Abbildung
P
:
F
k
→
[
0
,
1
]
{\displaystyle P:F^{k}\to [0,1]}
∫
R
k
f
(
x
)
d
k
x
=
1
{\displaystyle \int _{\mathbb {R} ^{k}}f(x)d^{k}x=1}
σ
{\displaystyle \sigma }
F
k
{\displaystyle F^{k}}
B
k
{\displaystyle {\mathcal {B}}^{k}}
b) Folgt dann aus dem vorangegangenen Satz und der Eindeutigkeitsaussage von a).
k
{\displaystyle k}
μ
∈
R
k
{\displaystyle \mu \in \mathbb {R} ^{k}}
Σ
{\displaystyle \Sigma }
k
×
k
{\displaystyle k\times k}
N
(
μ
,
Σ
)
{\displaystyle N(\mu ,\Sigma )}
Dichte:
f
(
x
)
=
1
(
s
π
)
k
d
e
t
(
Σ
)
e
−
1
2
(
x
−
μ
)
T
⋅
Σ
−
1
⋅
(
x
−
μ
)
{\displaystyle f(x)={\frac {1}{\sqrt {(s\pi )^{k}det(\Sigma )}}}e^{-{\frac {1}{2}}(x-\mu )^{T}\cdot \Sigma ^{-1}\cdot (x-\mu )}}
x
∈
R
k
.
{\displaystyle x\in \mathbb {R} ^{k}.}
Abkürzung:
(
N
k
(
μ
,
Σ
)
)
.
{\displaystyle (N_{k}(\mu ,\Sigma )).}
N
k
(
0
,
I
k
)
{\displaystyle N_{k}(0,I_{k})}
k
{\displaystyle k}
Im Fall
μ
=
0
∈
R
k
{\displaystyle \mu =0\in \mathbb {R} ^{k}}
Σ
=
I
k
{\displaystyle \Sigma =I_{k}}
k
{\displaystyle k}
f
(
x
)
=
1
(
s
π
)
k
e
−
1
2
(
x
1
2
+
.
.
.
+
x
k
2
)
=
Π
i
=
1
k
e
−
1
2
x
(
s
π
)
{\displaystyle f(x)={\frac {1}{\sqrt {(s\pi )^{k}}}}e^{-{\frac {1}{2}}(x_{1}^{2}+...+x_{k}^{2})}=\Pi _{i=1}^{k}{\frac {e^{-{\frac {1}{2}}x}}{\sqrt {(s\pi )}}}}
x
=
(
x
1
,
.
.
.
,
x
k
)
∈
R
k
{\displaystyle x=(x_{1},...,x_{k})\in \mathbb {R} ^{k}}
f
(
x
)
{\displaystyle f(x)}
Zu Beginn der Vorlesung hatten wir jede Abbildung:
Ω
→
Ω
′
{\displaystyle \Omega \to \Omega '}
(
Ω
,
S
,
P
)
→
(
Ω
′
,
S
′
,
P
X
)
{\displaystyle (\Omega ,{\mathcal {S}},P)\to (\Omega ',{\mathcal {S}}',P_{X})}
X
−
1
(
A
′
)
,
A
′
∈
S
′
{\displaystyle X^{-1}(A'),A'\in {\mathcal {S}}'}
S
{\displaystyle {\mathcal {S}}}
a) Sind
(
Ω
,
S
)
,
(
Ω
′
,
S
′
)
{\displaystyle (\Omega ,{\mathcal {S}}),(\Omega ',{\mathcal {S}}')}
X
:
Ω
→
Ω
′
{\displaystyle X:\Omega \to \Omega '}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
Ω
′
{\displaystyle \Omega '}
X
−
1
(
A
′
)
∈
S
∀
A
′
∈
S
′
{\displaystyle X^{-1}(A')\in {\mathcal {S}}\forall A'\in {\mathcal {S}}'}
b) Ist
X
:
Ω
→
Ω
′
{\displaystyle X:\Omega \to \Omega '}
P
{\displaystyle P}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
P
X
:
S
′
→
[
0
,
1
]
{\displaystyle P_{X}:{\mathcal {S}}'\to [0,1]}
P
X
(
A
′
)
=
P
(
X
−
1
(
A
′
)
)
,
A
′
∈
S
′
{\displaystyle P_{X}(A')=P(X^{-1}(A')),A'\in {\mathcal {S}}'}
Verteilung von
X
{\displaystyle X}
1. Man zeige genau wie zu Beginn der Vorlesung, dass
P
X
{\displaystyle P_{X}}
(
Ω
′
,
S
′
)
{\displaystyle (\Omega ',{\mathcal {S}}')}
2. In der Maßtheorie nennt man eine Abbildung
X
{\displaystyle X}
S
,
S
′
{\displaystyle {\mathcal {S}},{\mathcal {S}}'}
f
:
R
k
→
R
{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }
B
k
,
B
1
{\displaystyle {\mathcal {B}}^{k},{\mathcal {B}}^{1}}
3. Im Fall
Ω
′
=
R
k
,
S
′
=
B
k
{\displaystyle \Omega '=\mathbb {R} ^{k},{\mathcal {S}}'={\mathcal {B}}^{k}}
k
{\displaystyle k}
k
=
1
{\displaystyle k=1}
4. Es gibt nichtmessbare Funktionen
F
:
R
→
R
{\displaystyle F:\mathbb {R} \to \mathbb {R} }
C
⊂
R
1
{\displaystyle C\subset \mathbb {R} ^{1}}
f
=
1
C
{\displaystyle f=1_{C}}
Seien
(
Ω
,
S
)
,
(
Ω
′
,
S
′
)
{\displaystyle (\Omega ,{\mathcal {S}}),(\Omega ',{\mathcal {S}}')}
F
′
⊂
S
′
{\displaystyle F'\subset {\mathcal {S}}'}
S
′
{\displaystyle {\mathcal {S}}'}
σ
(
F
′
)
=
S
′
{\displaystyle \sigma (F')={\mathcal {S}}'}
X
:
Ω
→
Ω
′
{\displaystyle X:\Omega \to \Omega '}
X
−
1
(
A
′
)
∈
S
,
∀
A
′
∈
F
′
{\displaystyle X^{-1}(A')\in {\mathcal {S}},\forall A'\in F'}
Aus 1) folgt 3) (trivial). Sei nun 3) erfüllt. Setze
ϕ
′
=
{
A
′
∈
S
′
:
X
−
1
(
A
′
)
∈
S
}
{\displaystyle \phi '=\lbrace A'\in {\mathcal {S}}':X^{-1}(A')\in {\mathcal {S}}\rbrace }
man zeigt, dass
ϕ
′
{\displaystyle \phi '}
σ
{\displaystyle \sigma }
F
′
⊂
ϕ
′
⊂
S
′
{\displaystyle F'\subset \phi '\subset {\mathcal {S}}'}
S
′
=
σ
(
F
′
)
⊂
σ
(
ϕ
′
)
=
ϕ
′
⊂
σ
(
S
′
)
=
S
′
⇒
ϕ
′
=
S
′
.
{\displaystyle {\mathcal {S}}'=\sigma (F')\subset \sigma (\phi ')=\phi '\subset \sigma ({\mathcal {S}}')={\mathcal {S}}'\Rightarrow \phi '={\mathcal {S}}'.}
Sei
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
X
:
Ω
→
R
{\displaystyle X:\Omega \to \mathbb {R} }
{
X
<
b
}
≡
X
−
1
(
−
∞
,
b
]
∈
S
,
∀
b
∈
R
{\displaystyle \lbrace X<b\rbrace \equiv X^{-1}(-\infty ,b]\in {\mathcal {S}},\forall b\in \mathbb {R} }
(äquivalent:
≤
,
≥
,
>
{\displaystyle \leq ,\geq ,>}
<
{\displaystyle <}
X
:
R
k
→
R
{\displaystyle X:\mathbb {R} ^{k}\to \mathbb {R} }
(
R
k
,
B
k
)
{\displaystyle (\mathbb {R} ^{k},{\mathcal {B}}^{k})}
Setze
F
′
=
{
(
−
∞
,
b
]
:
b
∈
R
}
{\displaystyle F'=\lbrace (-\infty ,b]:b\in \mathbb {R} \rbrace }
σ
(
F
′
)
=
B
1
{\displaystyle \sigma (F')={\mathcal {B}}^{1}}
X
:
R
k
→
R
{\displaystyle X:\mathbb {R} ^{k}\to \mathbb {R} }
X
−
1
(
−
∞
,
b
)
{\displaystyle X^{-1}(-\infty ,b)}
R
k
{\displaystyle \mathbb {R} ^{k}}
B
k
{\displaystyle {\mathcal {B}}^{k}}
Sei
X
=
(
X
1
,
.
.
.
,
X
,
k
)
{\displaystyle X=(X_{1},...,X_{,}k)}
Ω
→
R
k
{\displaystyle \Omega \to \mathbb {R} ^{k}}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
X
{\displaystyle X}
X
i
{\displaystyle X_{i}}
i
=
1
,
.
.
.
,
k
{\displaystyle i=1,...,k}
Es gilt:
X
i
−
1
(
a
,
b
]
=
X
−
1
(
R
×
.
.
.
×
R
×
(
a
,
b
]
×
R
×
.
.
.
×
R
)
{\displaystyle X_{i}^{-1}(a,b]=X^{-1}(\mathbb {R} \times ...\times \mathbb {R} \times (a,b]\times \mathbb {R} \times ...\times \mathbb {R} )}
mit
(
a
,
b
]
{\displaystyle (a,b]}
i
{\displaystyle i}
Sind
(
Ω
,
S
)
,
(
Ω
′
,
S
′
)
,
(
Ω
″
,
S
″
)
{\displaystyle (\Omega ,{\mathcal {S}}),(\Omega ',{\mathcal {S}}'),(\Omega '',{\mathcal {S}}'')}
X
:
Ω
→
Ω
′
,
Y
:
Ω
′
→
Ω
″
{\displaystyle X:\Omega \to \Omega ',Y:\Omega '\to \Omega ''}
Y
∘
X
:
Ω
→
Ω
″
{\displaystyle Y\circ X:\Omega \to \Omega ''}
Die eingeführte Notaion "
F
{\displaystyle F}
P
{\displaystyle P}
f
{\displaystyle f}
P
{\displaystyle P}
P
X
{\displaystyle P_{X}}
X
{\displaystyle X}
Man sagt dann "
F
{\displaystyle F}
X
{\displaystyle X}
F
X
(
x
)
=
P
(
x
∈
X
)
=
P
X
(
−
∞
,
x
]
{\displaystyle F_{X}(x)=P(x\in X)=P_{X}(-\infty ,x]}
X
{\displaystyle X}
f
X
{\displaystyle f_{X}}
X
{\displaystyle X}
X
{\displaystyle X}
f
{\displaystyle f}
Ist die Zufallsvariable
X
{\displaystyle X}
P
X
{\displaystyle P_{X}}
λ
>
0
{\displaystyle \lambda >0}
X
{\displaystyle X}
Verteilungsfunktion:
F
X
(
x
)
=
{
0
,
x
<
0
1
−
e
−
λ
x
,
x
≥
0
{\displaystyle F_{X}(x)=\left\{{\begin{array}{ll}0,&x<0\\1-e^{-\lambda x},&x\geq 0\end{array}}\right.}
bzw. die Dichte:
f
X
(
x
)
=
{
0
,
x
<
0
λ
e
−
λ
x
,
x
≥
0
{\displaystyle f_{X}(x)=\left\{{\begin{array}{ll}0,&x<0\\\lambda e^{-\lambda x},&x\geq 0\end{array}}\right.}
Hat der Zufallsvektor
X
=
(
X
1
,
.
.
.
,
X
k
)
{\displaystyle X=(X_{1},...,X_{k})}
f
(
x
)
=
f
X
(
x
)
,
x
∈
R
k
{\displaystyle f(x)=f_{X}(x),x\in \mathbb {R} ^{k}}
k
{\displaystyle k}
(
a
,
b
]
=
⊗
i
=
1
k
(
a
i
,
b
i
]
{\displaystyle (a,b]=\otimes _{i=1}^{k}(a_{i},b_{i}]}
P
X
(
a
,
b
]
=
∫
a
k
b
k
.
.
.
∫
a
1
b
1
f
(
x
1
,
.
.
.
,
x
k
)
d
x
1
.
.
.
d
x
k
{\displaystyle P_{X}(a,b]=\int _{a_{k}}^{b_{k}}...\int _{a_{1}}^{b_{1}}f(x_{1},...,x_{k})dx_{1}...dx_{k}}
Hat
X
{\displaystyle X}
f
(
x
)
,
x
∈
R
k
{\displaystyle f(x),x\in \mathbb {R} ^{k}}
X
i
{\displaystyle X_{i}}
f
X
i
(
x
i
)
=
∫
R
k
−
1
f
(
x
1
,
.
.
.
,
x
k
)
d
x
1
.
.
.
d
x
i
−
1
d
x
i
+
1
.
.
.
d
x
k
{\displaystyle f_{X_{i}}(x_{i})=\int _{\mathbb {R} ^{k-1}}f(x_{1},...,x_{k})dx_{1}...dx_{i-1}dx_{i+1}...dx_{k}}
Der folgende Satz gibt die Dichte von
ϕ
∘
X
{\displaystyle \phi \circ X}
X
{\displaystyle X}
Der
k
{\displaystyle k}
X
{\displaystyle X}
f
(
x
)
,
x
∈
R
k
{\displaystyle f(x),x\in \mathbb {R} ^{k}}
U
⊂
R
k
{\displaystyle U\subset \mathbb {R} ^{k}}
f
(
x
)
=
0
{\displaystyle f(x)=0}
x
∉
U
{\displaystyle x\notin U}
ϕ
:
U
→
V
,
V
⊂
R
k
{\displaystyle \phi :U\to V,V\subset \mathbb {R} ^{k}}
ϕ
,
ϕ
−
1
{\displaystyle \phi ,\phi ^{-1}}
Dann hat der
k
{\displaystyle k}
Y
=
ϕ
∘
X
{\displaystyle Y=\phi \circ X}
g
(
y
)
=
0
,
y
∉
V
{\displaystyle g(y)=0,y\notin V}
g
(
y
)
=
f
(
ϕ
−
1
(
y
)
)
⋅
|
d
e
t
(
d
ϕ
−
1
d
y
(
y
)
)
|
,
y
∈
V
{\displaystyle g(y)=f(\phi ^{-1}(y))\cdot |det({\frac {d\phi ^{-1}}{dy}}(y))|,y\in V}
wobei
(
d
ϕ
−
1
d
y
(
y
)
)
=
(
d
ϕ
i
−
1
d
y
j
(
y
)
)
,
(
i
,
j
=
1
,
.
.
.
,
k
)
{\displaystyle ({\frac {d\phi ^{-1}}{dy}}(y))=({\frac {d\phi _{i}^{-1}}{dy_{j}}}(y)),(i,j=1,...,k)}
k
×
k
{\displaystyle k\times k}
ϕ
−
1
{\displaystyle \phi ^{-1}}
Zur Festlegung der Verteilung (und damit der Dichte) von
Y
=
ϕ
∘
X
{\displaystyle Y=\phi \circ X}
ϕ
{\displaystyle \phi }
U
{\displaystyle U}
ϕ
{\displaystyle \phi }
ϕ
~
:
R
k
→
R
k
{\displaystyle {\tilde {\phi }}:\mathbb {R} ^{k}\to \mathbb {R} ^{k}}
ϕ
|
U
=
ϕ
~
|
U
{\displaystyle \phi |U={\tilde {\phi }}|U}
P
ϕ
~
∘
X
=
P
ϕ
∘
X
{\displaystyle P_{{\tilde {\phi }}\circ X}=P_{\phi \circ X}}
In der Tat, sei
B
∈
B
k
{\displaystyle B\in {\mathcal {B}}^{k}}
P
ϕ
~
∘
X
(
B
)
=
P
(
X
∈
ϕ
~
−
1
(
B
)
)
=
P
(
X
∈
ϕ
~
−
1
(
B
)
∩
U
)
{\displaystyle P_{{\tilde {\phi }}\circ X}(B)=P(X\in {\tilde {\phi }}^{-1}(B))=P(X\in {\tilde {\phi }}^{-1}(B)\cap U)}
=
P
(
X
∈
ϕ
−
1
(
B
)
∩
U
)
=
P
(
X
∈
ϕ
−
1
(
B
)
)
=
P
ϕ
∘
X
(
B
)
{\displaystyle =P(X\in \phi ^{-1}(B)\cap U)=P(X\in \phi ^{-1}(B))=P_{\phi \circ X}(B)}
die zweite Gleichheit gilt wegen
P
(
X
∈
U
¯
)
=
∫
U
¯
f
(
x
)
d
x
=
0
,
{\displaystyle P(X\in {\bar {U}})=\int _{\bar {U}}f(x)dx=0,}
da
f
(
x
)
=
0
{\displaystyle f(x)=0}
x
∉
U
{\displaystyle x\notin U}
Sei
A
∈
B
k
{\displaystyle A\in {\mathcal {B}}^{k}}
P
(
X
∈
U
¯
)
=
0
{\displaystyle P(X\in {\bar {U}})=0}
P
ϕ
∘
X
(
A
)
=
P
(
X
∈
ϕ
−
1
(
A
)
)
=
P
(
X
∈
ϕ
−
1
(
A
)
∩
U
)
{\displaystyle P_{\phi \circ X}(A)=P(X\in \phi ^{-1}(A))=P(X\in \phi ^{-1}(A)\cap U)}
=
∫
ϕ
−
1
(
A
)
∩
U
f
(
x
)
d
x
=
∫
ϕ
−
1
(
a
∩
V
)
f
(
x
)
d
x
{\displaystyle =\int _{\phi ^{-1}(A)\cap U}f(x)dx=\int _{\phi ^{-1}(a\cap V)}f(x)dx}
=
∫
(
A
∩
V
)
f
(
ϕ
−
1
(
x
)
)
⋅
|
d
e
t
d
ϕ
−
1
d
y
(
y
)
|
d
y
≡
∫
A
g
(
y
)
d
y
{\displaystyle =\int _{(A\cap V)}f(\phi ^{-1}(x))\cdot |det{\frac {d\phi ^{-1}}{dy}}(y)|dy\equiv \int _{A}g(y)dy}
wobei wir den Transformationssatz für Integrale angewandt haben. Speziell gilt für offenes
A
=
⊗
i
=
k
k
(
a
i
,
b
i
)
{\displaystyle A=\otimes _{i=k}^{k}(a_{i},b_{i})}
P
ϕ
∘
X
(
A
)
=
∫
a
k
b
k
.
.
.
∫
a
1
b
1
g
(
y
)
d
y
{\displaystyle P_{\phi \circ X}(A)=\int _{a_{k}}^{b_{k}}...\int _{a_{1}}^{b_{1}}g(y)dy}
d.h.
g
(
y
)
{\displaystyle g(y)}
ϕ
∘
X
{\displaystyle \phi \circ X}
Besitzt der
k
{\displaystyle k}
X
{\displaystyle X}
f
(
x
)
,
x
∈
R
k
{\displaystyle f(x),x\in \mathbb {R} ^{k}}
g
(
y
)
,
y
∈
R
k
{\displaystyle g(y),y\in \mathbb {R} ^{k}}
Y
=
A
⋅
X
+
b
{\displaystyle Y=A\cdot X+b}
k
×
k
{\displaystyle k\times k}
b
∈
R
k
{\displaystyle b\in \mathbb {R} ^{k}}
g
(
y
)
=
1
|
d
e
t
A
|
f
(
A
−
1
(
y
−
b
)
)
,
y
∈
R
k
.
{\displaystyle g(y)={\frac {1}{|detA|}}f(A^{-1}(y-b)),y\in \mathbb {R} ^{k}.}
ϕ
(
X
)
=
A
⋅
X
+
b
{\displaystyle \phi (X)=A\cdot X+b}
U
=
R
k
{\displaystyle U=\mathbb {R} ^{k}}
ϕ
−
1
(
y
)
=
A
−
1
(
y
−
b
)
{\displaystyle \phi ^{-1}(y)=A^{-1}(y-b)}
d
e
t
(
d
ϕ
−
1
(
y
)
d
y
)
=
d
e
t
(
A
−
1
)
=
1
d
e
t
(
A
)
.
{\displaystyle det({\frac {d\phi ^{-1}(y)}{dy}})=det(A^{-1})={\frac {1}{det(A)}}.}
k
{\displaystyle k}
1. Ist
X
{\displaystyle X}
N
(
0
,
I
k
)
{\displaystyle N(0,I_{k})}
f
(
x
)
=
(
1
2
π
)
k
e
−
1
2
x
T
x
,
x
=
(
x
1
,
.
.
.
,
x
k
)
T
{\displaystyle f(x)=({\frac {1}{\sqrt {2\pi }}})^{k}e^{-{\frac {1}{2}}x^{T}x},x=(x_{1},...,x_{k})^{T}}
Y
=
A
⋅
X
+
μ
{\displaystyle Y=A\cdot X+\mu }
A
{\displaystyle A}
k
×
k
{\displaystyle k\times k}
μ
∈
R
k
{\displaystyle \mu \in \mathbb {R} ^{k}}
g
(
y
)
=
1
|
d
e
t
(
A
)
|
(
1
2
π
)
k
⋅
e
−
1
2
(
y
−
μ
)
T
(
A
−
1
)
T
⋅
A
−
1
(
y
−
μ
)
{\displaystyle g(y)={\frac {1}{|det(A)|}}({\frac {1}{\sqrt {2\pi }}})^{k}\cdot e^{-{\frac {1}{2}}(y-\mu )^{T}(A^{-1})^{T}\cdot A^{-1}(y-\mu )}}
=
1
(
2
π
)
k
d
e
t
(
Σ
)
e
−
1
2
(
y
−
μ
)
T
⋅
Σ
−
1
(
y
−
μ
)
{\displaystyle ={\frac {1}{\sqrt {(2\pi )^{k}det(\Sigma )}}}e^{-{\frac {1}{2}}(y-\mu )^{T}\cdot \Sigma ^{-1}(y-\mu )}}
mit
Σ
=
A
⋅
A
T
{\displaystyle \Sigma =A\cdot A^{T}}
Σ
−
1
=
(
A
⋅
A
T
)
−
1
=
(
A
−
1
)
⋅
A
−
1
{\displaystyle \Sigma ^{-1}=(A\cdot A^{T})^{-1}=(A^{-1})\cdot A^{-1}}
Σ
{\displaystyle \Sigma }
d
e
t
(
Σ
)
=
(
d
e
t
(
A
)
)
2
{\displaystyle det(\Sigma )=(det(A))^{2}}
Y
{\displaystyle Y}
N
k
(
μ
,
Σ
)
{\displaystyle N_{k}(\mu ,\Sigma )}
2. Ist umgekehrt
Y
N
k
(
μ
,
Σ
)
{\displaystyle YN_{k}(\mu ,\Sigma )}
Σ
{\displaystyle \Sigma }
X
=
(
Σ
−
1
2
)
T
⋅
(
y
−
μ
)
N
k
(
0
,
I
k
)
{\displaystyle X=(\Sigma ^{-{\frac {1}{2}}})^{T}\cdot (y-\mu )N_{k}(0,I_{k})}
Σ
−
1
2
{\displaystyle \Sigma ^{-{\frac {1}{2}}}}
k
×
k
{\displaystyle k\times k}
Σ
−
1
=
(
Σ
−
1
2
)
⋅
(
Σ
−
1
2
)
T
{\displaystyle \Sigma ^{-1}=(\Sigma ^{-{\frac {1}{2}}})\cdot (\Sigma ^{-{\frac {1}{2}}})^{T}}
(
Σ
−
1
2
)
T
=
A
{\displaystyle (\Sigma ^{-{\frac {1}{2}}})^{T}=A}
Für symmetrische, positiv definite
B
{\displaystyle B}
B
1
2
{\displaystyle B^{\frac {1}{2}}}
B
{\displaystyle B}
B
=
B
1
2
⋅
(
B
1
2
)
T
{\displaystyle B=B^{\frac {1}{2}}\cdot (B^{\frac {1}{2}})^{T}}
B
=
Σ
−
1
,
B
1
2
=
A
−
1
{\displaystyle B=\Sigma ^{-1},B^{\frac {1}{2}}=A^{-1}}
1. symmetrische Wurzel,
B
1
2
{\displaystyle B^{\frac {1}{2}}}
2. Cholesky Wurzel,
B
1
2
{\displaystyle B^{\frac {1}{2}}}
In jedem Fall ist
d
e
t
(
B
1
2
)
=
d
e
t
B
{\displaystyle det(B^{\frac {1}{2}})={\sqrt {detB}}}
a) Die auf
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
B
1
,
.
.
.
,
B
n
∈
B
1
{\displaystyle B_{1},...,B_{n}\in {\mathcal {B}}^{1}}
P
(
X
1
∈
B
1
,
.
.
.
,
X
n
∈
B
n
)
=
P
(
X
1
∈
B
1
)
⋅
.
.
.
⋅
P
(
X
n
∈
B
n
)
{\displaystyle P(X_{1}\in B_{1},...,X_{n}\in B_{n})=P(X_{1}\in B_{1})\cdot ...\cdot P(X_{n}\in B_{n})}
b) Abzählbar viele Zufallsvariablen
X
1
,
X
2
,
.
.
.
{\displaystyle X_{1},X_{2},...}
X
i
1
,
.
.
.
,
X
i
n
{\displaystyle X_{i_{1}},...,X_{i_{n}}}
Die auf
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
P
(
X
1
,
.
.
.
,
X
n
)
=
P
X
1
×
.
.
.
×
P
X
n
.
{\displaystyle P_{(X_{1},...,X_{n})}=P_{X_{1}}\times ...\times P_{X_{n}}.}
Es gelte
P
(
X
1
,
.
.
.
,
X
n
)
=
P
X
1
×
.
.
.
×
P
X
n
{\displaystyle P_{(X_{1},...,X_{n})}=P_{X_{1}}\times ...\times P_{X_{n}}}
Seien
B
1
,
.
.
,
B
n
∈
B
1
{\displaystyle B_{1},..,B_{n}\in {\mathcal {B}}^{1}}
P
(
X
1
∈
B
1
,
.
.
.
,
X
n
∈
B
n
)
=
P
(
X
1
,
.
.
.
,
X
n
)
(
B
1
×
.
.
.
×
B
n
)
{\displaystyle P(X_{1}\in B_{1},...,X_{n}\in B_{n})=P_{(X_{1},...,X_{n})}(B_{1}\times ...\times B_{n})}
=
P
X
1
×
.
.
.
×
P
X
n
(
B
1
×
.
.
.
×
B
n
)
=
P
X
1
(
B
1
)
⋅
.
.
.
⋅
P
X
n
(
B
n
)
{\displaystyle =P_{X_{1}}\times ...\times P_{X_{n}}(B_{1}\times ...\times B_{n})=P_{X_{1}}(B_{1})\cdot ...\cdot P_{X_{n}}(B_{n})}
=
P
(
X
1
∈
B
1
)
⋅
.
.
.
⋅
P
(
X
n
∈
B
n
)
{\displaystyle =P(X_{1}\in B_{1})\cdot ...\cdot P(X_{n}\in B_{n})}
Es gelte
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
P
(
X
1
,
.
.
.
,
X
n
)
(
B
1
×
.
.
.
×
B
n
)
=
P
(
X
1
⊂
B
1
)
⋅
.
.
.
⋅
P
(
X
n
⊂
B
n
)
{\displaystyle P_{(X_{1},...,X_{n})}(B_{1}\times ...\times B_{n})=P(X_{1}\subset B_{1})\cdot ...\cdot P(X_{n}\subset B_{n})}
=
P
(
X
1
∈
B
1
)
⋅
.
.
.
⋅
P
(
X
n
∈
B
n
)
=
P
X
1
(
B
1
)
⋅
.
.
.
⋅
P
X
n
(
B
n
)
{\displaystyle =P(X_{1}\in B_{1})\cdot ...\cdot P(X_{n}\in B_{n})=P_{X_{1}}(B_{1})\cdot ...\cdot P_{X_{n}}(B_{n})}
Da die
σ
{\displaystyle \sigma }
B
n
{\displaystyle {\mathcal {B}}^{n}}
(
a
,
b
]
=
⊗
1
n
(
a
i
,
b
i
]
{\displaystyle (a,b]=\otimes _{1}^{n}(a_{i},b_{i}]}
a
,
b
∈
R
n
,
a
<
b
{\displaystyle a,b\in \mathbb {R} ^{n},a<b}
P
(
a
1
⊂
X
1
≤
b
1
,
.
.
.
,
a
n
⊂
X
n
≤
b
n
)
=
P
(
a
1
⊂
X
1
≤
b
1
)
⋅
.
.
.
⋅
P
(
a
n
⊂
X
n
≤
b
n
)
{\displaystyle P(a_{1}\subset X_{1}\leq b_{1},...,a_{n}\subset X_{n}\leq b_{n})=P(a_{1}\subset X_{1}\leq b_{1})\cdot ...\cdot P(a_{n}\subset X_{n}\leq b_{n})}
Auch Intervalltypen
[
)
,
[
]
,
(
)
{\displaystyle [),[],()}
Die Zufallsvariablen
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
f
1
,
.
.
.
,
f
n
{\displaystyle f_{1},...,f_{n}}
Dann gilt:
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
⇑⇓
{\displaystyle \Uparrow \Downarrow }
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle (X_{1},...,X_{n})}
f
(
x
1
,
.
.
.
,
x
n
)
=
f
(
x
1
)
⋅
.
.
.
⋅
f
(
x
n
)
{\displaystyle f(x_{1},...,x_{n})=f(x_{1})\cdot ...\cdot f(x_{n})}
"
⇓
{\displaystyle \Downarrow }
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
P
(
x
1
,
.
.
.
,
x
n
)
(
(
a
1
,
b
1
]
×
.
.
.
×
(
a
n
,
b
n
]
)
=
P
X
1
(
a
1
,
b
1
]
⋅
.
.
.
⋅
P
X
n
(
a
n
,
b
n
]
{\displaystyle P_{(x_{1},...,x_{n})}((a_{1},b_{1}]\times ...\times (a_{n},b_{n}])=P_{X_{1}}(a_{1},b_{1}]\cdot ...\cdot P_{X_{n}}(a_{n},b_{n}]}
=
∫
a
1
b
1
f
1
(
x
1
)
d
x
1
⋅
.
.
.
⋅
∫
a
n
b
n
f
n
(
x
n
)
d
x
n
{\displaystyle =\int _{a_{1}}^{b_{1}}f_{1}(x_{1})dx_{1}\cdot ...\cdot \int _{a_{n}}^{b_{n}}f_{n}(x_{n})dx_{n}}
=
∫
a
1
b
1
.
.
.
∫
a
n
b
n
f
1
(
x
1
)
⋅
.
.
.
⋅
f
n
(
x
n
)
d
x
1
⋅
.
.
.
⋅
d
x
n
{\displaystyle =\int _{a_{1}}^{b_{1}}...\int _{a_{n}}^{b_{n}}f_{1}(x_{1})\cdot ...\cdot f_{n}(x_{n})dx_{1}\cdot ...\cdot dx_{n}}
⇒
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle \Rightarrow (X_{1},...,X_{n})}
f
(
x
1
,
.
.
.
,
x
n
)
,
(
x
1
,
.
.
.
,
x
n
)
∈
R
n
{\displaystyle f(x_{1},...,x_{n}),(x_{1},...,x_{n})\in \mathbb {R} ^{n}}
"
⇑
{\displaystyle \Uparrow }
X
=
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle X=(X_{1},...,X_{n})}
N
(
0
,
I
k
)
{\displaystyle {\mathcal {N}}(0,I_{k})}
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
N
(
0
,
1
)
{\displaystyle {\mathcal {N}}(0,1)}
Sind
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
f
X
i
(
x
)
=
f
i
(
x
)
=
1
2
π
e
−
x
2
2
,
x
∈
R
{\displaystyle f_{X_{i}}(x)=f_{i}(x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}},x\in \mathbb {R} }
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle (X_{1},...,X_{n})}
f
(
x
1
,
.
.
.
,
x
n
)
=
Π
i
=
1
n
f
i
(
x
i
)
=
1
2
π
2
e
−
1
2
(
x
1
2
+
.
.
.
+
x
n
2
)
=
1
2
π
n
e
−
1
2
(
x
T
⋅
x
)
{\displaystyle f(x_{1},...,x_{n})=\Pi _{i=1}^{n}f_{i}(x_{i})={\frac {1}{{\sqrt {2\pi }}^{2}}}e^{-{\frac {1}{2}}(x_{1}^{2}+...+x_{n}^{2})}={\frac {1}{{\sqrt {2\pi }}^{n}}}e^{-{\frac {1}{2}}(x^{T}\cdot x)}}
mit
x
=
(
x
1
,
.
.
.
,
x
n
)
{\displaystyle x=(x_{1},...,x_{n})}
X
{\displaystyle X}
f
(
x
1
,
.
.
.
,
x
n
)
=
1
2
π
n
e
−
1
2
(
x
T
⋅
x
)
=
Π
i
=
1
n
1
2
π
n
e
−
1
2
(
x
i
2
)
{\displaystyle f(x_{1},...,x_{n})={\frac {1}{{\sqrt {2\pi }}n}}e^{-{\frac {1}{2}}(x^{T}\cdot x)}=\Pi _{i=1}^{n}{\frac {1}{{\sqrt {2\pi }}^{n}}}e^{-{\frac {1}{2}}(x_{i}^{2})}}
Man stellt fest, durch Integration
∫
−
∞
∞
{\displaystyle \int _{-\infty }^{\infty }}
x
1
,
.
.
.
,
x
i
−
1
,
x
i
+
1
,
.
.
.
,
x
n
{\displaystyle x_{1},...,x_{i-1},x_{i+1},...,x_{n}}
f
i
(
x
i
)
{\displaystyle f_{i}(x_{i})}
X
i
{\displaystyle X_{i}}
N
(
0
,
1
)
{\displaystyle {\mathcal {N}}(0,1)}
X
i
{\displaystyle X_{i}}
Für unabhängige
X
1
,
X
2
{\displaystyle X_{1},X_{2}}
P
X
1
+
X
2
(
{
x
1
}
)
=
∑
k
P
X
1
(
{
x
1
}
)
⋅
P
X
2
(
{
x
−
x
1
}
)
{\displaystyle P_{X_{1}+X_{2}}(\lbrace x_{1}\rbrace )=\sum _{k}P_{X_{1}}(\lbrace x_{1}\rbrace )\cdot P_{X_{2}}(\lbrace x-x_{1}\rbrace )}
Sind
X
1
,
X
2
{\displaystyle X_{1},X_{2}}
f
1
,
f
2
{\displaystyle f_{1},f_{2}}
X
1
+
X
2
{\displaystyle X_{1}+X_{2}}
f
(
x
)
=
∫
−
∞
∞
f
1
(
x
1
)
f
2
(
x
−
x
1
)
d
x
.
{\displaystyle f(x)=\int _{-\infty }^{\infty }f_{1}(x_{1})f_{2}(x-x_{1})dx.}
Für die Verteilungsfunktion
F
(
y
)
:
P
(
X
1
+
X
2
≤
y
)
{\displaystyle F(y):P(X_{1}+X_{2}\leq y)}
F
(
y
)
=
∫
−
∞
y
f
(
x
)
d
x
{\displaystyle F(y)=\int _{-\infty }^{y}f(x)dx}
P
(
X
1
+
X
2
≤
y
)
=
P
(
X
1
,
X
2
)
(
{
(
x
1
,
x
2
)
:
x
1
+
x
2
≤
y
}
)
{\displaystyle P(X_{1}+X_{2}\leq y)=P_{(X_{1},X_{2})}(\lbrace (x_{1},x_{2}):x_{1}+x_{2}\leq y\rbrace )}
=
∫
{
(
x
1
,
x
2
)
:
x
1
+
x
2
≤
y
}
f
1
(
x
1
)
⋅
f
2
(
x
2
)
d
x
2
d
x
1
{\displaystyle =\int _{\lbrace (x_{1},x_{2}):x_{1}+x_{2}\leq y\rbrace }f_{1}(x_{1})\cdot f_{2}(x_{2})dx_{2}dx_{1}}
=
∫
−
∞
∞
∫
−
∞
y
−
x
1
f
1
(
x
1
)
⋅
f
2
(
x
2
)
d
x
2
d
x
1
{\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{y-x_{1}}f_{1}(x_{1})\cdot f_{2}(x_{2})dx_{2}dx_{1}}
=
∫
−
∞
∞
∫
−
∞
y
f
1
(
x
1
)
⋅
f
2
(
x
2
−
x
1
)
d
x
2
d
x
1
{\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{y}f_{1}(x_{1})\cdot f_{2}(x_{2}-x_{1})dx_{2}dx_{1}}
=
∫
−
∞
y
∫
−
∞
∞
f
1
(
x
1
)
⋅
f
2
(
x
2
−
x
)
d
x
1
d
x
2
{\displaystyle =\int _{-\infty }^{y}\int _{-\infty }^{\infty }f_{1}(x_{1})\cdot f_{2}(x_{2}-x)dx_{1}dx_{2}}
=
∫
−
∞
y
∫
−
∞
∞
f
1
(
x
1
)
⋅
f
2
(
x
−
x
1
)
d
x
1
d
x
2
{\displaystyle =\int _{-\infty }^{y}\int _{-\infty }^{\infty }f_{1}(x_{1})\cdot f_{2}(x-x_{1})dx_{1}dx_{2}}
mit der Dichte
p
(
x
)
=
∫
−
∞
∞
f
1
(
x
1
)
⋅
f
2
(
x
−
x
1
)
d
x
1
.
{\displaystyle p(x)=\int _{-\infty }^{\infty }f_{1}(x_{1})\cdot f_{2}(x-x_{1})dx_{1}.}
Sind
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
P
X
1
+
.
.
.
+
X
n
≡
P
X
1
∗
.
.
.
∗
P
X
n
{\displaystyle P_{X_{1}+...+X_{n}}\equiv P_{X_{1}}*...*P_{X_{n}}}
Faltung von
P
X
1
+
.
.
.
+
X
n
{\displaystyle P_{X_{1}+...+X_{n}}}
∗
{\displaystyle *}
Sei
S
n
{\displaystyle S_{n}}
n
−
1
{\displaystyle n-1}
n
{\displaystyle n}
T
n
=
S
1
+
.
.
.
+
S
n
{\displaystyle T_{n}=S_{1}+...+S_{n}}
n
{\displaystyle n}
1. Die Zufallsvariablen
S
1
,
S
2
,
.
.
.
{\displaystyle S_{1},S_{2},...}
2. Jedes
S
i
{\displaystyle S_{i}}
λ
{\displaystyle \lambda }
ϵ
(
λ
)
{\displaystyle \epsilon (\lambda )}
wollen wir die Dichte der Zufallsvariable
T
n
{\displaystyle T_{n}}
f
n
(
x
)
=
λ
n
⋅
x
n
−
1
(
n
−
1
)
!
⋅
e
−
λ
x
,
x
≥
0
{\displaystyle f_{n}(x)={\frac {\lambda ^{n}\cdot x^{n-1}}{(n-1)!}}\cdot e^{-\lambda x},x\geq 0}
f
n
(
x
)
=
0
{\displaystyle f_{n}(x)=0}
x
<
0
{\displaystyle x<0}
Die Wahrscheinlichkeitsverteilung mit der Dichte
f
n
{\displaystyle f_{n}}
n
{\displaystyle n}
λ
{\displaystyle \lambda }
Γ
(
n
,
λ
)
{\displaystyle \Gamma (n,\lambda )}
I
n
{\displaystyle I_{n}}
Γ
(
n
,
λ
)
{\displaystyle \Gamma (n,\lambda )}
Γ
(
n
,
λ
)
=
ϵ
(
λ
)
∗
.
.
.
∗
ϵ
(
λ
)
{\displaystyle \Gamma (n,\lambda )=\epsilon (\lambda )*...*\epsilon (\lambda )}
Zerlegt man einen Satz von Zufallsvariablen in disjunkte Gruppen und setzt auf die Gruppen Funktionen an, so erhalten wir unabhängige Zufallsvariablen.
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
m
≤
n
{\displaystyle m\leq n}
{
1
,
.
.
.
,
n
}
=
I
1
∪
.
.
.
∪
I
m
{\displaystyle \lbrace 1,...,n\rbrace =I_{1}\cup ...\cup I_{m}}
ϕ
j
{\displaystyle \phi _{j}}
(
R
k
j
,
b
k
j
)
{\displaystyle (\mathbb {R} ^{k_{j}},{\mathcal {b}}^{k_{j}})}
k
j
=
|
I
j
|
{\displaystyle k_{j}=|I_{j}|}
j
=
1
,
.
.
.
,
m
{\displaystyle j=1,...,m}
(
∑
j
=
1
m
k
j
=
n
)
{\displaystyle (\sum _{j=1}^{m}k_{j}=n)}
Y
j
{\displaystyle Y_{j}}
k
j
{\displaystyle k_{j}}
(
X
i
,
i
∈
I
)
{\displaystyle (X_{i},i\in I)}
ϕ
1
⋅
Y
1
,
.
.
.
,
ϕ
m
⋅
Y
m
{\displaystyle \phi _{1}\cdot Y_{1},...,\phi _{m}\cdot Y_{m}}
unabhängige Zufallsvariablen.
Ohne Einschränkung sei
I
1
=
{
1
,
.
.
.
,
k
1
}
,
I
2
=
{
k
1
+
1
,
.
.
.
,
k
1
+
k
2
}
,
.
.
.
{\displaystyle I_{1}=\lbrace 1,...,k_{1}\rbrace ,I_{2}=\lbrace k_{1}+1,...,k_{1}+k_{2}\rbrace ,...}
Zunächst zeigen wir, dass die
m
{\displaystyle m}
Y
1
,
.
.
.
,
Y
m
{\displaystyle Y_{1},...,Y_{m}}
(*)
P
(
Y
1
,
.
.
.
,
Y
m
)
(
C
1
×
.
.
.
×
C
m
)
=
P
Y
1
(
C
1
)
⋅
.
.
.
⋅
P
Y
m
(
C
m
)
{\displaystyle P_{(Y_{1},...,Y_{m})}(C_{1}\times ...\times C_{m})=P_{Y_{1}}(C_{1})\cdot ...\cdot P_{Y_{m}}(C_{m})}
für alle
C
j
∈
B
k
j
,
j
=
1
,
.
.
.
,
m
{\displaystyle C_{j}\in {\mathcal {B}}^{k_{j}},j=1,...,m}
Für die speziellen
C
j
{\displaystyle C_{j}}
C
j
=
B
1
j
×
.
.
.
×
B
k
j
,
B
l
j
∈
B
1
{\displaystyle C_{j}=B_{1}^{j}\times ...\times B_{k}^{j},B_{l}^{j}\in {\mathcal {B}}^{1}}
(
Y
1
,
.
.
.
,
Y
m
)
=
(
X
1
,
.
.
.
,
X
n
)
,
(
C
1
,
.
.
.
,
C
m
)
=
(
B
1
j
,
.
.
.
,
B
k
m
j
)
{\displaystyle (Y_{1},...,Y_{m})=(X_{1},...,X_{n}),(C_{1},...,C_{m})=(B_{1}^{j},...,B_{k_{m}}^{j})}
P
(
Y
1
,
.
.
.
,
Y
m
)
(
C
1
×
.
.
.
×
C
m
)
=
P
(
X
1
,
.
.
.
,
X
n
)
(
B
1
1
×
.
.
.
×
B
k
m
m
)
{\displaystyle P_{(Y_{1},...,Y_{m})}(C_{1}\times ...\times C_{m})=P_{(X_{1},...,X_{n})}(B_{1}^{1}\times ...\times B_{k_{m}}^{m})}
=
P
X
1
(
B
1
1
)
⋅
.
.
.
⋅
P
X
n
(
B
k
m
m
)
{\displaystyle =P_{X_{1}}(B_{1}^{1})\cdot ...\cdot P_{X_{n}}(B_{k_{m}}^{m})}
=
⊗
i
∈
I
1
P
X
i
(
B
1
1
×
.
.
.
×
B
k
1
1
)
⋅
.
.
.
⋅
⊗
i
∈
I
m
P
X
i
(
B
1
m
×
.
.
.
×
B
k
m
m
)
{\displaystyle =\otimes _{i\in I_{1}}P_{X_{i}}(B_{1}^{1}\times ...\times B_{k_{1}}^{1})\cdot ...\cdot \otimes _{i\in I_{m}}P_{X_{i}}(B_{1}^{m}\times ...\times B_{k_{m}}^{m})}
=
P
Y
1
(
C
1
)
⋅
.
.
.
⋅
P
Y
m
(
C
m
)
{\displaystyle =P_{Y_{1}}(C_{1})\cdot ...\cdot P_{Y_{m}}(C_{m})}
Nach dem Fortsetzungssatz gilt dann (*) auch für alle
C
j
∈
B
k
j
{\displaystyle C_{j}\in {\mathcal {B}}^{k_{j}}}
Nun wird die Unabhängigkeit der
ϕ
1
∘
Y
1
,
.
.
.
,
ϕ
m
∘
Y
m
{\displaystyle \phi _{1}\circ Y_{1},...,\phi _{m}\circ Y_{m}}
P
ϕ
1
∘
Y
1
,
.
.
.
,
ϕ
m
∘
Y
m
)
(
B
1
×
.
.
.
×
B
m
)
=
.
.
.
=
P
ϕ
1
∘
Y
1
(
B
1
)
⋅
.
.
.
⋅
P
ϕ
m
∘
Y
m
(
B
m
)
{\displaystyle P_{\phi _{1}\circ Y_{1},...,\phi _{m}\circ Y_{m})}(B_{1}\times ...\times B_{m})=...=P_{\phi _{1}\circ Y_{1}}(B_{1})\cdot ...\cdot P_{\phi _{m}\circ Y_{m}}(B_{m})}
Wir führen den Begriff des Erwartungswert
E
(
X
)
{\displaystyle E(X)}
X
{\displaystyle X}
X
{\displaystyle X}
X
n
{\displaystyle X_{n}}
Für eine beliebige Zufallsvariable
X
{\displaystyle X}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
n
∈
N
{\displaystyle n\in \mathbb {N} }
n
{\displaystyle n}
X
n
(
w
)
=
∑
i
=
−
∞
∞
k
n
1
A
k
,
n
(
w
)
{\displaystyle X_{n}(w)=\sum _{i=-\infty }^{\infty }{\frac {k}{n}}1_{A_{k,n}}(w)}
A
k
,
n
=
{
w
:
k
n
≤
X
(
w
)
<
k
+
1
n
}
{\displaystyle A_{k,n}=\lbrace w:{\frac {k}{n}}\leq X(w)<{\frac {k+1}{n}}\rbrace }
d.h.
{
X
n
(
w
)
k
n
,
w
∈
A
k
,
n
0
,
s
o
n
s
t
{\displaystyle \left\{{\begin{array}{ll}X_{n}(w){\frac {k}{n}},&w\in A_{k,n}\\0,&sonst\end{array}}\right.}
Es ist
A
k
,
n
∈
S
{\displaystyle A_{k,n}\in {\mathcal {S}}}
X
n
{\displaystyle X_{n}}
±
k
n
,
k
∈
Z
{\displaystyle \pm {\frac {k}{n}},k\in \mathbb {Z} }
X
n
{\displaystyle X_{n}}
E
(
X
n
)
=
∑
i
=
−
∞
∞
P
(
A
k
,
n
)
{\displaystyle E(X_{n})=\sum _{i=-\infty }^{\infty }P(A_{k,n})}
(mit
P
(
A
k
,
n
)
=
P
X
n
{
k
n
}
{\displaystyle P(A_{k,n})=P_{X_{n}}\lbrace {\frac {k}{n}}\rbrace }
∑
i
=
−
∞
∞
|
k
|
n
P
(
A
k
,
n
)
≡
E
(
|
X
n
|
)
<
∞
.
{\displaystyle \sum _{i=-\infty }^{\infty }{\frac {|k|}{n}}P(A_{k,n})\equiv E(|X_{n}|)<\infty .}
a)
X
n
≤
X
≤
X
n
+
1
n
{\displaystyle X_{n}\leq X\leq X_{n}+{\frac {1}{n}}}
|
X
−
X
n
|
≤
1
n
{\displaystyle |X-X_{n}|\leq {\frac {1}{n}}}
b)
|
X
n
−
X
m
|
≤
1
n
+
1
m
{\displaystyle |X_{n}-X_{m}|\leq {\frac {1}{n}}+{\frac {1}{m}}}
|
X
n
−
X
m
≤
|
X
n
−
X
|
+
|
X
−
X
m
|
{\displaystyle |X_{n}-X_{m}\leq |X_{n}-X|+|X-X_{m}|}
c)
E
|
X
n
−
X
m
|
≤
1
n
+
1
m
{\displaystyle E|X_{n}-X_{m}|\leq {\frac {1}{n}}+{\frac {1}{m}}}
E
{\displaystyle E}
d) Existiert
E
(
X
n
)
{\displaystyle E(X_{n})}
n
∈
N
{\displaystyle n\in \mathbb {N} }
E
(
X
m
)
{\displaystyle E(X_{m})}
m
≥
n
{\displaystyle m\geq n}
E
(
X
m
)
≤
E
|
X
m
−
X
n
|
+
E
(
X
m
)
<
∞
{\displaystyle E(X_{m})\leq E|X_{m}-X_{n}|+E(X_{m})<\infty }
e) Existiert
E
(
X
n
)
{\displaystyle E(X_{n})}
n
∈
N
{\displaystyle n\in \mathbb {N} }
E
(
X
n
)
,
n
≥
n
0
{\displaystyle E(X_{n}),n\geq n_{0}}
|
E
(
X
n
)
−
E
(
X
m
)
|
=
|
E
(
X
n
−
X
m
)
|
≤
E
|
X
n
−
X
m
|
=
1
n
+
1
m
→
0
,
n
,
m
→
∞
{\displaystyle |E(X_{n})-E(X_{m})|=|E(X_{n}-X_{m})|\leq E|X_{n}-X_{m}|={\frac {1}{n}}+{\frac {1}{m}}\to 0,n,m\to \infty }
Falls für (mindestens) ein
n
∈
N
{\displaystyle n\in \mathbb {N} }
E
(
X
n
)
{\displaystyle E(X_{n})}
n
{\displaystyle n}
X
{\displaystyle X}
E
(
X
)
=
l
i
m
n
→
∞
E
(
X
n
)
{\displaystyle E(X)=lim_{n\to \infty }E(X_{n})}
E
(
X
)
{\displaystyle E(X)}
X
{\displaystyle X}
E
(
X
)
=
∫
a
X
(
w
)
P
(
d
w
)
{\displaystyle E(X)=\int _{a}X(w)P(dw)}
Dieses "
P
{\displaystyle P}
X
{\displaystyle X}
y
{\displaystyle y}
x
{\displaystyle x}
a)
E
(
X
)
{\displaystyle E(X)}
E
|
X
|
{\displaystyle E|X|}
E
|
X
|
<
∞
{\displaystyle E|X|<\infty }
b) Ist
X
(
Ω
)
{\displaystyle X(\Omega )}
E
(
X
)
=
∑
x
∈
X
(
Ω
)
x
⋅
P
X
{
w
}
{\displaystyle E(X)=\sum _{x\in X(\Omega )}x\cdot P_{X}\lbrace w\rbrace }
a) Mehrfache Anwendung der Eigenschaften von
E
(
X
n
)
{\displaystyle E(X_{n})}
|
X
n
|
≤
|
X
|
+
|
X
−
X
n
|
≤
|
X
|
+
1
n
≤
|
X
|
n
+
2
n
{\displaystyle |X_{n}|\leq |X|+|X-X_{n}|\leq |X|+{\frac {1}{n}}\leq |X|_{n}+{\frac {2}{n}}}
|
X
|
n
≤
.
.
.
≤
|
X
n
|
+
1
n
{\displaystyle |X|_{n}\leq ...\leq |X_{n}|+{\frac {1}{n}}}
b) Setze
I
k
,
n
=
(
k
n
,
k
+
1
n
]
{\displaystyle I_{k,n}=({\frac {k}{n}},{\frac {k+1}{n}}]}
P
(
X
n
=
k
n
)
=
∑
x
∈
I
k
,
n
P
(
X
=
x
)
{\displaystyle P(X_{n}={\frac {k}{n}})=\sum _{x\in I_{k,n}}P(X=x)}
(*)
{
E
(
X
n
)
=
∑
k
=
−
∞
∞
k
n
∑
x
∈
I
k
,
n
P
(
X
=
x
)
≤
∑
k
=
−
∞
∞
∑
x
∈
I
k
,
n
x
P
(
X
=
x
)
=
∑
x
∈
X
(
Ω
)
x
P
(
X
=
x
)
≤
∑
k
=
−
∞
∞
k
+
1
n
∑
x
∈
T
k
,
n
P
(
X
=
x
)
=
E
(
X
n
)
+
1
n
{\displaystyle \left\{{\begin{array}{ll}E(X_{n})=\sum _{k=-\infty }^{\infty }{\frac {k}{n}}\sum _{x\in I_{k,n}}P(X=x)\\\leq \sum _{k=-\infty }^{\infty }\sum _{x\in I_{k,n}}xP(X=x)\\=\sum _{x\in X(\Omega )}xP(X=x)\\\leq \sum _{k=-\infty }^{\infty }{\frac {k+1}{n}}\sum _{x\in T_{k,n}}P(X=x)=E(X_{n})+{\frac {1}{n}}\end{array}}\right.}
Falls die Reihe
∑
x
P
(
X
=
x
)
{\displaystyle \sum xP(X=x)}
E
(
X
n
)
≤
∑
|
x
|
P
(
X
=
x
)
+
1
n
<
∞
{\displaystyle E(X_{n})\leq \sum |x|P(X=x)+{\frac {1}{n}}<\infty }
E
(
X
n
)
{\displaystyle E(X_{n})}
l
i
m
n
→
∞
{\displaystyle lim_{n\to \infty }}
Im speziellen Fall, dass
X
{\displaystyle X}
E
(
X
)
{\displaystyle E(X)}
Besitzt die Zufallsvariable
X
{\displaystyle X}
f
(
x
)
,
x
∈
R
{\displaystyle f(x),x\in \mathbb {R} }
E
(
X
)
=
∫
−
∞
∞
x
f
(
x
)
d
x
,
{\displaystyle E(X)=\int _{-\infty }^{\infty }xf(x)dx,}
sofern
∫
−
∞
∞
|
x
|
f
(
x
)
d
x
<
∞
.
{\displaystyle \int _{-\infty }^{\infty }|x|f(x)dx<\infty .}
Wegen
P
(
X
n
=
k
n
)
=
P
(
k
n
≤
X
≤
k
+
1
n
)
=
∫
k
n
k
+
1
n
f
(
x
)
d
x
{\displaystyle P(X_{n}={\frac {k}{n}})=P({\frac {k}{n}}\leq X\leq {\frac {k+1}{n}})=\int _{\frac {k}{n}}^{\frac {k+1}{n}}f(x)dx}
(*)
E
(
X
n
)
=
∑
k
=
−
∞
∞
k
n
∫
k
n
k
+
1
n
f
(
x
)
d
x
≤
∑
k
=
−
∞
∞
x
f
(
x
)
d
x
{\displaystyle E(X_{n})=\sum _{k=-\infty }^{\infty }{\frac {k}{n}}\int _{\frac {k}{n}}^{\frac {k+1}{n}}f(x)dx\leq \sum _{k=-\infty }^{\infty }xf(x)dx}
=
∫
−
∞
∞
x
f
(
x
)
d
x
≤
∑
k
=
−
∞
∞
k
+
1
n
∫
k
n
k
+
1
n
x
f
(
x
)
d
x
=
E
(
X
n
)
+
1
n
{\displaystyle =\int _{-\infty }^{\infty }xf(x)dx\leq \sum _{k=-\infty }^{\infty }{\frac {k+1}{n}}\int _{\frac {k}{n}}^{\frac {k+1}{n}}xf(x)dx=E(X_{n})+{\frac {1}{n}}}
(Ähnliche Überlegung zur absoluten Konvergenz
E
(
X
n
)
l
i
m
n
→
∞
{\displaystyle E(X_{n})lim_{n\to \infty }}
Allgemeiner gilt der folgende Satz (
k
{\displaystyle k}
X
{\displaystyle X}
ϕ
∘
X
{\displaystyle \phi \circ X}
Besitzt ein
k
{\displaystyle k}
X
{\displaystyle X}
f
(
x
)
,
x
∈
R
{\displaystyle f(x),x\in \mathbb {R} }
ϕ
{\displaystyle \phi }
R
k
→
R
{\displaystyle \mathbb {R} ^{k}\to \mathbb {R} }
E
(
ϕ
∘
X
)
=
∫
R
k
ϕ
(
x
)
f
(
x
)
d
k
x
,
{\displaystyle E(\phi \circ X)=\int _{\mathbb {R} ^{k}}\phi (x)f(x)d^{k}x,}
sofern
∫
R
k
|
ϕ
|
f
d
k
x
<
∞
.
{\displaystyle \int _{\mathbb {R} ^{k}}|\phi |fd^{k}x<\infty .}
Gemäß dem Satz über Verkettung von Zufallsvariablen ist
ϕ
∘
X
{\displaystyle \phi \circ X}
E
(
ϕ
∘
X
)
n
=
.
.
.
≤
.
.
.
=
∫
R
k
ϕ
(
x
)
f
(
x
)
d
k
x
≤
.
.
.
=
E
(
ϕ
∘
X
)
n
+
1
n
{\displaystyle E(\phi \circ X)_{n}=...\leq ...=\int _{\mathbb {R} ^{k}}\phi (x)f(x)d^{k}x\leq ...=E(\phi \circ X)_{n}+{\frac {1}{n}}}
Wie bei diskreten Zufallsvariablen haben wir auch hier die Monotonie und die Linearität des Erwartungswertes.
Sind
X
{\displaystyle X}
Y
{\displaystyle Y}
E
(
X
)
{\displaystyle E(X)}
E
(
Y
)
{\displaystyle E(Y)}
a)
E
(
a
X
+
b
Y
)
{\displaystyle E(aX+bY)}
E
(
a
X
+
b
Y
)
=
a
E
(
X
)
+
b
E
(
Y
)
{\displaystyle E(aX+bY)=aE(X)+bE(Y)}
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
b)
E
(
X
)
≤
E
(
Y
)
{\displaystyle E(X)\leq E(Y)}
X
≤
Y
{\displaystyle X\leq Y}
c)
E
(
1
)
=
1
{\displaystyle E(1)=1}
Folgt aus den entsprechenden Eigenschaften für diskrete Zufallsvariablen. Für die Existenz des Erwartungswertes ist das sogenannte Majorantenkriterium nützlich.
Sind
X
,
Y
{\displaystyle X,Y}
|
X
|
≤
Y
{\displaystyle |X|\leq Y}
E
(
Y
)
{\displaystyle E(Y)}
E
(
Y
)
<
∞
{\displaystyle E(Y)<\infty }
E
(
X
)
{\displaystyle E(X)}
E
(
X
)
≤
E
(
Y
)
{\displaystyle E(X)\leq E(Y)}
Für die approximierten Zufallsvariablen
|
X
|
n
{\displaystyle |X|_{n}}
Y
n
{\displaystyle Y_{n}}
|
X
|
n
≤
Y
n
{\displaystyle |X|_{n}\leq Y_{n}}
E
(
|
X
|
n
)
≤
E
(
Y
n
)
<
∞
{\displaystyle E(|X|_{n})\leq E(Y_{n})<\infty }
(Letzteres für
n
≤
n
0
{\displaystyle n\leq n_{0}}
E
|
X
|
{\displaystyle E|X|}
E
(
X
)
{\displaystyle E(X)}
E
(
X
)
{\displaystyle E(X)}
Existieren für unabhängige Zufallsvariablen
X
{\displaystyle X}
Y
{\displaystyle Y}
E
(
X
)
{\displaystyle E(X)}
E
(
Y
)
{\displaystyle E(Y)}
X
⋅
Y
{\displaystyle X\cdot Y}
E
(
X
⋅
Y
)
=
E
(
X
)
⋅
E
(
Y
)
.
{\displaystyle E(X\cdot Y)=E(X)\cdot E(Y).}
Man kann die Approximation
X
n
,
Y
n
{\displaystyle X_{n},Y_{n}}
X
n
=
ϕ
(
X
)
,
Y
n
=
ϕ
(
Y
)
{\displaystyle X_{n}=\phi (X),Y_{n}=\phi (Y)}
ϕ
≡
ϕ
n
{\displaystyle \phi \equiv \phi _{n}}
X
n
,
Y
n
{\displaystyle X_{n},Y_{n}}
X
n
⋅
Y
n
{\displaystyle X_{n}\cdot Y_{n}}
E
(
X
n
⋅
Y
n
)
=
E
(
X
n
)
⋅
E
(
Y
n
)
.
{\displaystyle E(X_{n}\cdot Y_{n})=E(X_{n})\cdot E(Y_{n}).}
Wir haben die Ungleichung
|
(
X
⋅
Y
)
n
−
X
n
⋅
Y
n
|
{\displaystyle |(X\cdot Y)_{n}-X_{n}\cdot Y_{n}|}
≤
|
(
X
⋅
Y
)
n
−
|
X
⋅
Y
|
+
|
X
⋅
Y
|
−
|
X
n
⋅
Y
|
+
|
X
n
⋅
Y
|
−
X
n
⋅
Y
n
|
{\displaystyle \leq |(X\cdot Y)_{n}-|X\cdot Y|+|X\cdot Y|-|X_{n}\cdot Y|+|X_{n}\cdot Y|-X_{n}\cdot Y_{n}|}
≤
1
n
+
1
n
|
Y
|
+
1
n
|
X
n
|
≤
1
n
(
2
+
|
X
|
+
|
Y
|
)
≡
1
n
Z
{\displaystyle \leq {\frac {1}{n}}+{\frac {1}{n}}|Y|+{\frac {1}{n}}|X_{n}|\leq {\frac {1}{n}}(2+|X|+|Y|)\equiv {\frac {1}{n}}Z}
E
(
X
⋅
Y
)
n
{\displaystyle E(X\cdot Y)_{n}}
E
(
X
⋅
Y
)
{\displaystyle E(X\cdot Y)}
E
(
X
n
⋅
Y
n
)
−
E
(
X
⋅
Y
)
n
→
n
→
∞
0
{\displaystyle E(X_{n}\cdot Y_{n})-E(X\cdot Y)_{n}\to ^{n\to \infty }0}
Für das nun folgende ('höhere Momente') wird wiederholt folgende Ungleichung benutzt:
|
a
±
b
|
m
≤
C
m
(
|
a
|
m
+
|
b
|
m
)
{\displaystyle |a\pm b|^{m}\leq C_{m}(|a|^{m}+|b|^{m})}
für alle
a
,
b
∈
R
,
m
∈
N
{\displaystyle a,b\in \mathbb {R} ,m\in \mathbb {N} }
C
m
=
2
m
−
1
{\displaystyle C_{m}=2^{m-1}}
Diese Ungleichung folgt aus der Jensenschen Ungleichung in der Form (
r
,
m
∈
N
,
a
i
>
0
{\displaystyle r,m\in \mathbb {N} ,a_{i}>0}
1
r
m
(
a
1
+
.
.
.
+
a
r
)
m
≤
1
r
(
a
1
m
+
.
.
.
+
a
r
m
)
{\displaystyle {\frac {1}{r^{m}}}(a_{1}+...+a_{r})^{m}\leq {\frac {1}{r}}(a_{1}^{m}+...+a_{r}^{m})}
(im Beweis ist
r
=
2
{\displaystyle r=2}
Sei
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
a) Für
m
∈
N
{\displaystyle m\in \mathbb {N} }
L
m
≡
L
m
(
P
)
{\displaystyle {\mathcal {L}}_{m}\equiv {\mathcal {L}}_{m}(P)}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
E
|
X
|
m
<
∞
{\displaystyle E|X|^{m}<\infty }
X
∈
L
m
{\displaystyle X\in {\mathcal {L}}_{m}}
E
|
X
|
m
{\displaystyle E|X|^{m}}
m
{\displaystyle m}
E
(
X
m
)
{\displaystyle E(X^{m})}
m
{\displaystyle m}
b) Für
X
∈
L
m
{\displaystyle X\in {\mathcal {L}}_{m}}
m
{\displaystyle m}
E
(
(
X
−
E
X
)
m
)
{\displaystyle E((X-EX)^{m})}
m
{\displaystyle m}
E
(
|
X
E
X
|
m
)
{\displaystyle E(|X_{E}X|^{m})}
c) Speziell für
X
∈
L
2
{\displaystyle X\in {\mathcal {L}}_{2}}
V
a
r
(
X
)
=
E
(
X
E
X
)
2
{\displaystyle Var(X)=E(X_{E}X)^{2}}
X
{\displaystyle X}
σ
(
X
)
=
V
a
r
(
X
)
{\displaystyle \sigma (X)={\sqrt {Var(X)}}}
X
{\displaystyle X}
V
a
r
(
a
X
+
b
)
=
a
2
V
a
r
(
X
)
{\displaystyle Var(aX+b)=a^{2}Var(X)}
V
a
r
(
X
)
=
E
(
X
2
−
(
E
X
)
2
)
{\displaystyle Var(X)=E(X^{2}-(EX)^{2})}
Ferner gilt:
V
a
r
(
X
)
=
0
{\displaystyle Var(X)=0}
P
(
X
=
c
o
n
s
t
)
=
1
{\displaystyle P(X=const)=1}
X
=
c
o
n
s
t
.
{\displaystyle X=const.}
P
{\displaystyle P}
E
X
2
=
0
{\displaystyle EX^{2}=0}
P
(
X
=
0
)
=
1
{\displaystyle P(X=0)=1}
X
=
0
{\displaystyle X=0}
P
{\displaystyle P}
X
{\displaystyle X}
[
a
,
b
]
{\displaystyle [a,b]}
a
<
b
{\displaystyle a<b}
X
∗
=
X
−
a
b
−
a
{\displaystyle X^{*}={\frac {X-a}{b-a}}}
[
0
,
1
]
{\displaystyle [0,1]}
E
X
∗
=
∫
0
1
x
⋅
1
⋅
d
x
=
1
2
↦
X
=
a
+
(
b
−
a
)
X
∗
E
X
=
a
⋅
1
2
(
b
−
a
)
=
1
2
(
a
+
b
)
{\displaystyle EX^{*}=\int _{0}^{1}x\cdot 1\cdot dx={\frac {1}{2}}{\stackrel {X=a+(b-a)X^{*}}{\mapsto }}EX=a\cdot {\frac {1}{2}}(b-a)={\frac {1}{2}}(a+b)}
E
(
X
∗
)
=
∫
0
1
x
⋅
1
⋅
d
x
=
1
3
{\displaystyle E(X^{*})=\int _{0}^{1}x\cdot 1\cdot dx={\frac {1}{3}}}
V
a
r
(
X
∗
)
=
1
3
+
1
4
=
1
12
,
{\displaystyle Var(X^{*})={\frac {1}{3}}+{\frac {1}{4}}={\frac {1}{12}},}
also
V
a
r
(
X
)
=
1
12
(
b
−
a
)
2
{\displaystyle Var(X)={\frac {1}{12}}(b-a)^{2}}
X
{\displaystyle X}
λ
>
0
{\displaystyle \lambda >0}
E
X
=
∫
0
∞
x
⋅
λ
⋅
e
−
λ
x
⋅
d
x
=
x
=
λ
x
1
λ
∫
0
∞
s
⋅
e
−
x
⋅
d
s
=
1
λ
{\displaystyle EX=\int _{0}^{\infty }x\cdot \lambda \cdot e^{-\lambda x}\cdot dx{\stackrel {x=\lambda x}{=}}{\frac {1}{\lambda }}\int _{0}^{\infty }s\cdot e^{-x}\cdot ds={\frac {1}{\lambda }}}
E
X
2
=
∫
0
∞
x
2
⋅
λ
⋅
e
−
λ
x
=
s
=
λ
x
1
λ
2
∫
0
∞
s
2
⋅
e
−
s
⋅
d
s
=
2
λ
2
{\displaystyle EX^{2}=\int _{0}^{\infty }x^{2}\cdot \lambda \cdot e^{-\lambda x}{\stackrel {s=\lambda x}{=}}{\frac {1}{\lambda ^{2}}}\int _{0}^{\infty }s^{2}\cdot e^{-s}\cdot ds={\frac {2}{\lambda ^{2}}}}
V
a
r
(
X
)
=
2
λ
2
−
1
λ
2
=
1
λ
2
{\displaystyle Var(X)={\frac {2}{\lambda ^{2}}}-{\frac {1}{\lambda ^{2}}}={\frac {1}{\lambda ^{2}}}}
Normalverteilung
N
(
μ
,
σ
2
)
{\displaystyle N(\mu ,\sigma ^{2})}
Ist
X
{\displaystyle X}
N
(
μ
,
σ
)
{\displaystyle N(\mu ,\sigma )}
X
∗
=
X
−
μ
σ
{\displaystyle X^{*}={\frac {X-\mu }{\sigma }}}
N
(
0
,
1
)
{\displaystyle N(0,1)}
E
X
∗
=
∫
−
∞
∞
|
x
>
e
−
1
2
x
2
d
x
=
0
{\displaystyle EX^{*}=\int _{-\infty }^{\infty }|x>e^{-{\frac {1}{2}}x^{2}}dx=0}
wegen
ϕ
(
x
)
=
ϕ
(
−
x
)
{\displaystyle \phi (x)=\phi (-x)}
∫
∞
∞
|
x
>
e
1
2
x
2
d
x
<
∞
.
{\displaystyle \int _{\infty }^{\infty }|x>e^{\frac {1}{2x^{2}}}dx<\infty .}
Ferner:
V
a
r
(
X
∗
)
=
E
(
X
∗
)
2
=
∫
−
∞
∞
x
⋅
x
ϕ
(
x
)
d
x
=
∫
−
∞
∞
ϕ
(
x
)
d
x
=
1
{\displaystyle Var(X^{*})=E(X^{*})^{2}=\int _{-\infty }^{\infty }x\cdot x\phi (x)dx=\int _{-\infty }^{\infty }\phi (x)dx=1}
Es folgt für
X
=
μ
+
σ
X
∗
{\displaystyle X=\mu +\sigma X^{*}}
E
X
=
μ
{\displaystyle EX=\mu }
V
a
r
(
X
)
=
σ
2
.
{\displaystyle Var(X)=\sigma ^{2}.}
Die
N
(
μ
,
σ
2
)
{\displaystyle N(\mu ,\sigma ^{2})}
μ
{\displaystyle \mu }
σ
2
{\displaystyle \sigma ^{2}}
Den Anschluss an die Lineare Algebra/Funktionalanalysis liefert der folgende Satz.
Seien
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
m
∈
N
{\displaystyle m\in \mathbb {N} }
a)
L
m
{\displaystyle {\mathcal {L}}_{m}}
b)
L
n
⊂
L
m
{\displaystyle {\mathcal {L}}_{n}\subset {\mathcal {L}}_{m}}
n
≥
m
{\displaystyle n\geq m}
E
|
X
|
n
<
∞
{\displaystyle E|X|^{n}<\infty }
n
∈
N
{\displaystyle n\in \mathbb {N} }
E
|
X
|
m
<
∞
{\displaystyle E|X|^{m}<\infty }
m
≤
n
{\displaystyle m\leq n}
E
|
X
|
<
∞
{\displaystyle E|X|<\infty }
a) Majorantenkriterium und die Ungleichung des letzten Satzes liefern für
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
|
a
X
+
b
Y
|
m
≤
C
m
(
|
a
|
m
|
X
|
m
+
|
b
|
m
|
Y
|
m
)
{\displaystyle |aX+bY|^{m}\leq C_{m}(|a|^{m}|X|^{m}+|b|^{m}|Y|^{m})}
b) Sei
E
|
X
|
n
<
∞
{\displaystyle E|X|^{n}<\infty }
m
≤
n
{\displaystyle m\leq n}
|
X
|
m
≤
1
⋅
1
{
|
X
|
≤
1
}
+
|
X
|
1
{
|
X
|
≥
1
}
≤
1
+
|
X
|
n
{\displaystyle |X|^{m}\leq 1\cdot 1_{\lbrace |X|\leq 1\rbrace }+|X|1_{\lbrace |X|\geq 1\rbrace }\leq 1+|X|^{n}}
E
|
X
|
m
<
∞
{\displaystyle E|X|^{m}<\infty }
Wichtig sind die folgenden stochastischen Ungleichungen.
Markov-Ungleichung:
Ist
X
∈
L
m
{\displaystyle X\in {\mathcal {L}}_{m}}
m
∈
N
{\displaystyle m\in \mathbb {N} }
ϵ
<
0
{\displaystyle \epsilon <0}
P
(
|
X
|
≥
ϵ
)
≤
E
|
X
|
m
ϵ
m
{\displaystyle P(|X|\geq \epsilon )\leq {\frac {E|X|^{m}}{\epsilon ^{m}}}}
Tschebyschoff-Ungleichung:
Insbesondere für
X
∈
L
2
{\displaystyle X\in {\mathcal {L}}_{2}}
P
(
|
X
−
E
X
|
≥
ϵ
)
≤
V
a
r
(
X
)
ϵ
2
{\displaystyle P(|X-EX|\geq \epsilon )\leq {\frac {Var(X)}{\epsilon ^{2}}}}
Wiederholte Anwendung der Monotonieeigenschaften von
E
{\displaystyle E}
E
|
X
|
m
≥
E
(
|
X
|
m
1
{
|
X
|
≥
1
}
)
≥
ϵ
m
E
(
1
{
|
X
|
≥
1
}
)
=
ϵ
P
(
|
X
|
≥
ϵ
)
{\displaystyle E|X|^{m}\geq E(|X|^{m}1_{\lbrace |X|\geq 1\rbrace })\geq \epsilon ^{m}E(1_{\lbrace |X|\geq 1\rbrace })=\epsilon P(|X|\geq \epsilon )}
setzt man in die Markov-Ungleichung speziell
X
−
E
X
{\displaystyle X-EX}
X
{\displaystyle X}
m
=
2
{\displaystyle m=2}
Für Zufallsvariablen
X
,
Y
∈
L
2
{\displaystyle X,Y\in {\mathcal {L}}_{2}}
X
⋅
Y
∈
L
1
{\displaystyle X\cdot Y\in {\mathcal {L}}_{1}}
[
E
(
X
⋅
Y
)
]
2
≤
E
(
X
)
2
⋅
E
(
Y
)
2
{\displaystyle [E(X\cdot Y)]^{2}\leq E(X)^{2}\cdot E(Y)^{2}}
a
X
+
b
Y
=
0
{\displaystyle aX+bY=0}
P
{\displaystyle P}
a
,
b
,
a
2
+
b
2
>
0
{\displaystyle a,b,a^{2}+b^{2}>0}
Im linearen Raum
L
2
{\displaystyle {\mathcal {L}}_{2}}
Für
X
,
Y
∈
L
1
{\displaystyle X,Y\in {\mathcal {L}}_{1}}
⟨
X
,
Y
⟩
=
E
(
X
,
Y
)
{\displaystyle \langle X,Y\rangle =E(X,Y)}
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
⟨
X
,
X
⟩
≥
0
{\displaystyle \langle X,X\rangle \geq 0}
⟨
X
,
X
⟩
=
0
{\displaystyle \langle X,X\rangle =0}
X
=
0
{\displaystyle X=0}
X
=
0
{\displaystyle X=0}
Sind
X
,
Y
∈
L
2
{\displaystyle X,Y\in {\mathcal {L}}_{2}}
a)
C
o
v
(
X
,
Y
)
=
E
(
(
X
−
E
X
)
⋅
(
Y
−
E
Y
)
)
=
E
(
X
⋅
Y
)
−
E
(
X
)
⋅
E
(
Y
)
{\displaystyle Cov(X,Y)=E((X-EX)\cdot (Y-EY))=E(X\cdot Y)-E(X)\cdot E(Y)}
X
{\displaystyle X}
Y
{\displaystyle Y}
b)
X
,
Y
{\displaystyle X,Y}
C
o
v
(
X
,
Y
)
=
0
{\displaystyle Cov(X,Y)=0}
c)
ρ
(
X
,
Y
)
=
C
o
v
(
X
,
Y
)
σ
(
X
)
σ
(
Y
)
{\displaystyle \rho (X,Y)={\frac {Cov(X,Y)}{\sigma (X)\sigma (Y)}}}
X
{\displaystyle X}
Y
{\displaystyle Y}
σ
(
X
)
,
σ
(
Y
)
>
0
{\displaystyle \sigma (X),\sigma (Y)>0}
Die Folgerungen für diskrete Zufallsvariablen bezüglich der Kovarianz gelten weiterhin sowie die Eigenschaften von der Varianz und der Kovarianz. Im Hinblick auf die obige Bemerkung gilt:
X
,
Y
{\displaystyle X,Y}
X
−
E
X
⊥
Y
−
E
Y
{\displaystyle X-EX\perp Y-EY}
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
Momente der
k
{\displaystyle k}
Ist
X
=
(
X
1
,
.
.
.
,
X
k
)
{\displaystyle X=(X_{1},...,X_{k})}
N
k
(
μ
,
Σ
)
{\displaystyle N_{k}(\mu ,\Sigma )}
μ
=
(
μ
1
,
.
.
.
,
μ
k
)
T
∈
R
k
,
Σ
=
(
σ
i
,
j
)
{\displaystyle \mu =(\mu _{1},...,\mu _{k})^{T}\in \mathbb {R} ^{k},\Sigma =(\sigma _{i,j})}
k
×
k
{\displaystyle k\times k}
Behauptung:
E
X
i
=
μ
i
,
C
o
v
(
X
i
,
X
j
)
=
σ
i
,
j
{\displaystyle EX_{i}=\mu _{i},Cov(X_{i},X_{j})=\sigma _{i,j}}
Die Parameter
μ
,
Σ
{\displaystyle \mu ,\Sigma }
N
k
(
μ
,
Σ
)
{\displaystyle N_{k}(\mu ,\Sigma )}
N
k
(
μ
,
Σ
)
{\displaystyle N_{k}(\mu ,\Sigma )}
X
{\displaystyle X}
Für diskrete Zufallsvariablen
X
{\displaystyle X}
Z
+
{\displaystyle \mathbb {Z} _{+}}
G
(
s
)
=
E
s
X
=
∑
k
=
0
∞
s
k
P
X
{
k
}
{\displaystyle G(s)=Es^{X}=\sum _{k=0}^{\infty }s^{k}P_{X}\lbrace k\rbrace }
Eine vergleichbare Funktion hat die charakteristische Funktion in der allgemeinen Wahrscheinlichkeitstheorie, in der
X
{\displaystyle X}
s
X
{\displaystyle s^{X}}
e
i
X
{\displaystyle e^{iX}}
Für eine komplexe Zahl
z
=
a
+
b
i
,
a
=
R
e
(
z
)
∈
R
,
b
=
I
m
(
z
)
∈
R
{\displaystyle z=a+bi,a=Re(z)\in \mathbb {R} ,b=Im(z)\in \mathbb {R} }
|
z
|
=
a
2
+
b
2
=
z
⋅
z
¯
{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}={\sqrt {z\cdot {\bar {z}}}}}
z
=
r
⋅
e
i
ϕ
{\displaystyle z=r\cdot e^{i\phi }}
r
=
|
z
|
,
e
i
ϕ
=
c
o
s
(
ϕ
)
+
i
sin
(
ϕ
)
{\displaystyle r=|z|,e^{i\phi }=cos(\phi )+i\sin(\phi )}
|
z
⋅
w
|
=
|
z
|
⋅
|
w
|
{\displaystyle |z\cdot w|=|z|\cdot |w|}
Sei
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
a) Sind
z
1
,
z
2
{\displaystyle z_{1},z_{2}}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
z
i
:
Ω
→
R
{\displaystyle z_{i}:\Omega \to \mathbb {R} }
z
=
z
1
+
i
z
2
{\displaystyle z=z_{1}+iz_{2}}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
z
:
Ω
→
C
{\displaystyle z:\Omega \to \mathbb {C} }
b) Existieren
E
(
z
1
)
,
E
(
z
2
)
{\displaystyle E(z_{1}),E(z_{2})}
E
(
z
)
:=
E
(
z
1
)
+
i
E
(
z
2
)
{\displaystyle E(z):=E(z_{1})+iE(z_{2})}
z
{\displaystyle z}
a) Sind
z
,
z
¯
{\displaystyle z,{\bar {z}}}
E
(
z
)
,
E
(
z
¯
)
{\displaystyle E(z),E({\bar {z}})}
E
(
z
+
z
¯
)
=
E
(
z
)
+
E
(
z
¯
)
{\displaystyle E(z+{\bar {z}})=E(z)+E({\bar {z}})}
E
(
v
⋅
z
)
=
v
⋅
E
(
z
)
,
v
∈
C
{\displaystyle E(v\cdot z)=v\cdot E(z),v\in \mathbb {C} }
b)
|
E
(
z
)
|
≤
E
(
|
z
|
)
<
∞
{\displaystyle |E(z)|\leq E(|z|)<\infty }
Bearbeiten
Sei
X
{\displaystyle X}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
ϕ
X
:
R
→
C
{\displaystyle \phi _{X}:\mathbb {R} \to \mathbb {C} }
ϕ
X
(
t
)
≡
ϕ
(
t
)
=
e
i
t
x
=
E
(
c
o
s
(
t
x
)
)
+
i
E
(
s
i
n
(
t
x
)
)
{\displaystyle \phi _{X}(t)\equiv \phi (t)=e^{itx}=E(cos(tx))+iE(sin(tx))}
charakteristische Funktion von
X
{\displaystyle X}
Aus
e
i
t
x
=
c
o
s
(
t
x
)
+
i
s
i
n
(
t
x
)
{\displaystyle e^{itx}=cos(tx)+isin(tx)}
|
c
o
s
(
t
x
)
|
≤
1
{\displaystyle |cos(tx)|\leq 1}
|
s
i
n
(
t
x
)
|
≤
1
{\displaystyle |sin(tx)|\leq 1}
E
(
c
o
s
(
t
x
)
)
{\displaystyle E(cos(tx))}
E
(
s
i
n
(
t
x
)
)
{\displaystyle E(sin(tx))}
(
e
i
x
t
)
{\displaystyle (e^{ixt})}
Beispiele für charakteristische Funktionen:
ϕ
X
(
t
)
=
1
{\displaystyle \phi _{X}(t)=1}
ϕ
X
(
t
)
=
c
o
s
(
t
)
{\displaystyle \phi _{X}(t)=cos(t)}
ϕ
X
(
t
)
=
e
−
t
2
2
{\displaystyle \phi _{X}(t)=e^{-{\frac {t^{2}}{2}}}}
ϕ
X
(
t
)
=
e
e
i
t
−
1
{\displaystyle \phi _{X}(t)=e^{e^{it}-1}}
ϕ
X
(
t
)
=
1
1
+
i
t
{\displaystyle \phi _{X}(t)={\frac {1}{1+it}}}
ϕ
X
(
t
)
=
1
i
t
(
e
i
t
−
1
)
,
(
ϕ
X
(
0
)
=
1
{\displaystyle \phi _{X}(t)={\frac {1}{it}}(e^{it}-1),(\phi _{X}(0)=1}
ϕ
X
(
t
)
=
e
−
|
t
|
{\displaystyle \phi _{X}(t)=e^{-|t|}}
ϕ
X
(
t
)
=
(
1
−
|
t
|
)
1
[
−
1
,
1
]
(
t
)
{\displaystyle \phi _{X}(t)=(1-|t|)1_{[-1,1]}(t)}
Keine charakteristischen Funktionen sind:
ϕ
(
t
)
=
s
i
n
(
t
)
{\displaystyle \phi (t)=sin(t)}
ϕ
(
t
)
=
1
−
t
2
{\displaystyle \phi (t)=1-t^{2}}
ϕ
(
t
)
=
1
[
−
1
,
1
]
(
t
)
{\displaystyle \phi (t)=1_{[-1,1]}(t)}
ϕ
(
t
)
=
e
−
|
t
|
2
{\displaystyle \phi (t)=e^{-|t|^{2}}}
Wegen
|
e
i
t
x
|
=
1
{\displaystyle |e^{itx}|=1}
|
ϕ
(
t
)
|
=
|
E
(
e
i
t
x
)
|
≤
E
(
|
e
i
x
t
|
)
=
E
(
1
)
=
1
{\displaystyle |\phi (t)|=|E(e^{itx})|\leq E(|e^{ixt}|)=E(1)=1}
ϕ
(
0
)
=
E
(
1
)
=
1
{\displaystyle \phi (0)=E(1)=1}
ϕ
X
{\displaystyle \phi _{X}}
ϕ
a
X
+
b
(
t
)
=
E
(
e
i
t
(
a
X
+
b
)
)
=
e
i
t
b
⋅
E
(
e
i
t
a
X
)
=
i
t
b
⋅
ϕ
X
(
t
a
)
,
a
,
b
∈
R
.
{\displaystyle \phi _{aX+b}(t)=E(e^{it(aX+b)})=e^{itb}\cdot E(e^{itaX})=itb\cdot \phi _{X}(ta),a,b\in \mathbb {R} .}
Ist
X
{\displaystyle X}
Z
+
{\displaystyle \mathbb {Z} _{+}}
ϕ
X
(
t
)
=
E
(
e
i
t
x
)
=
E
(
c
o
s
(
t
x
)
)
+
i
E
(
s
i
n
(
t
x
)
)
{\displaystyle \phi _{X}(t)=E(e^{itx})=E(cos(tx))+iE(sin(tx))}
=
∑
k
=
0
∞
c
o
s
(
t
k
)
⋅
P
(
X
=
k
)
+
i
∑
k
=
0
∞
s
i
n
(
t
k
)
⋅
P
(
X
=
k
)
{\displaystyle =\sum _{k=0}^{\infty }cos(tk)\cdot P(X=k)+i\sum _{k=0}^{\infty }sin(tk)\cdot P(X=k)}
=
∑
k
=
0
∞
(
e
i
t
)
k
⋅
P
(
{
k
}
)
{\displaystyle =\sum _{k=0}^{\infty }(e^{it})^{k}\cdot P(\lbrace k\rbrace )}
(vgl. mit
g
x
(
s
)
=
E
(
s
x
)
=
∑
k
=
k
∞
s
k
⋅
P
x
(
{
k
}
)
,
s
∈
[
0
,
1
]
{\displaystyle g_{x}(s)=E(s^{x})=\sum _{k=k}^{\infty }sk\cdot P_{x}(\lbrace k\rbrace ),s\in [0,1]}
X
{\displaystyle X}
X
{\displaystyle X}
B
(
n
,
p
)
{\displaystyle B(n,p)}
ϕ
X
(
t
)
=
(
1
−
p
+
p
e
i
t
)
n
,
t
∈
R
{\displaystyle \phi _{X}(t)=(1-p+pe^{it})^{n},t\in \mathbb {R} }
X
{\displaystyle X}
P
(
λ
)
{\displaystyle P(\lambda )}
ϕ
X
(
t
)
=
e
λ
e
i
t
−
1
,
t
∈
R
{\displaystyle \phi _{X}(t)=e^{\lambda e^{it}-1},t\in \mathbb {R} }
X
{\displaystyle X}
λ
>
0
{\displaystyle \lambda >0}
ϕ
X
(
t
)
=
λ
λ
−
i
t
,
t
∈
R
{\displaystyle \phi _{X}(t)={\frac {\lambda }{\lambda -it}},t\in \mathbb {R} }
X
{\displaystyle X}
N
(
0
,
1
)
{\displaystyle N(0,1)}
ϕ
X
(
t
)
=
E
(
e
i
t
x
)
=
E
(
c
o
s
(
t
x
)
)
+
i
E
(
s
i
n
(
t
x
)
)
{\displaystyle \phi _{X}(t)=E(e^{itx})=E(cos(tx))+iE(sin(tx))}
=
∫
−
∞
∞
(
c
o
s
(
t
x
)
)
1
2
π
e
−
x
2
2
d
x
+
i
∫
−
∞
∞
(
s
i
n
(
t
x
)
)
1
2
π
e
−
x
2
2
d
x
{\displaystyle =\int _{-\infty }^{\infty }(cos(tx)){\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}dx+i\int _{-\infty }^{\infty }(sin(tx)){\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}dx}
⇒
ϕ
X
′
(
t
)
=
∫
−
∞
∞
(
s
i
n
(
t
x
)
)
1
2
π
(
−
x
)
e
−
x
2
2
d
x
{\displaystyle \Rightarrow \phi '_{X}(t)=\int _{-\infty }^{\infty }(sin(tx)){\frac {1}{\sqrt {2\pi }}}(-x)e^{-{\frac {x^{2}}{2}}}dx}
=
[
(
s
i
n
(
t
x
)
)
1
2
π
(
−
x
)
e
−
x
2
2
]
−
∞
∞
{\displaystyle =[(sin(tx)){\frac {1}{\sqrt {2\pi }}}(-x)e^{-{\frac {x^{2}}{2}}}]_{-\infty }^{\infty }}
=
∫
−
∞
∞
t
⋅
c
o
s
(
t
x
)
1
2
π
e
−
x
2
2
d
x
{\displaystyle =\int _{-\infty }^{\infty }t\cdot cos(tx){\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}dx}
=
−
t
⋅
ϕ
X
(
t
)
d
d
t
(
ϕ
X
(
t
)
⋅
e
−
x
2
2
)
{\displaystyle =-t\cdot \phi _{X}(t){\frac {d}{dt}}(\phi _{X}(t)\cdot e^{-{\frac {x^{2}}{2}}})}
=
ϕ
X
′
(
t
)
⋅
e
−
x
2
2
+
ϕ
X
(
t
)
⋅
t
⋅
e
−
x
2
2
{\displaystyle =\phi '_{X}(t)\cdot e^{-{\frac {x^{2}}{2}}}+\phi _{X}(t)\cdot t\cdot e^{-{\frac {x^{2}}{2}}}}
=
e
−
x
2
2
[
ϕ
X
′
(
t
)
+
t
ϕ
X
(
t
)
]
=
0
{\displaystyle =e^{-{\frac {x^{2}}{2}}}[\phi '_{X}(t)+t\phi _{X}(t)]=0}
⇒
ϕ
X
(
t
)
⋅
e
−
x
2
2
=
c
=
c
o
n
s
t
.
{\displaystyle \Rightarrow \phi _{X}(t)\cdot e^{-{\frac {x^{2}}{2}}}=c=const.}
⇒
{
ϕ
X
(
t
)
=
c
⋅
e
−
x
2
2
ϕ
X
(
0
)
=
1
⇒
c
=
1
}
{\displaystyle \Rightarrow \left\{{\begin{array}{ll}\phi _{X}(t)=c\cdot e^{-{\frac {x^{2}}{2}}}\\\phi _{X}(0)=1\Rightarrow c=1\end{array}}\right\}}
⇒
ϕ
X
(
t
)
=
e
−
x
2
2
{\displaystyle \Rightarrow \phi _{X}(t)=e^{-{\frac {x^{2}}{2}}}}
X
{\displaystyle X}
N
(
0
,
1
)
{\displaystyle N(0,1)}
⇒
ϕ
X
(
t
)
=
?
{\displaystyle \Rightarrow \phi _{X}(t)=?}
X
=
σ
Γ
+
μ
,
Γ
{\displaystyle X=\sigma \Gamma +\mu ,\Gamma }
N
(
0
,
1
)
{\displaystyle N(0,1)}
⇒
ϕ
X
(
t
)
⋅
e
i
t
μ
⋅
ϕ
Y
(
σ
t
)
=
e
i
t
μ
⋅
e
−
1
2
σ
2
t
2
.
{\displaystyle \Rightarrow \phi _{X}(t)\cdot e^{it\mu }\cdot \phi _{Y}(\sigma t)=e^{it\mu }\cdot e^{-{\frac {1}{2}}\sigma ^{2}t^{2}}.}
Seien
X
,
Y
{\displaystyle X,Y}
ϕ
X
=
ϕ
Y
⇔
P
X
=
P
Y
{\displaystyle \phi _{X}=\phi _{Y}\Leftrightarrow P_{X}=P_{Y}}
Sind
X
,
Y
{\displaystyle X,Y}
ϕ
X
+
Y
=
ϕ
X
+
ϕ
y
{\displaystyle \phi _{X+Y}=\phi _{X}+\phi _{y}}
ϕ
X
+
Y
=
E
(
e
i
t
(
x
+
y
)
)
=
E
(
e
i
t
x
⋅
e
i
t
y
)
=
E
(
e
i
t
x
)
⋅
E
(
e
i
t
y
)
=
ϕ
X
+
ϕ
Y
{\displaystyle \phi _{X+Y}=E(e^{it(x+y)})=E(e^{itx}\cdot e^{ity})=E(e^{itx})\cdot E(e^{ity})=\phi _{X}+\phi _{Y}}
Für den obigen Beweis wurde der folgende Hilfssatz genutzt.
Seien
X
,
Y
{\displaystyle X,Y}
f
=
f
1
+
f
2
{\displaystyle f=f_{1}+f_{2}}
g
=
g
1
+
g
2
{\displaystyle g=g_{1}+g_{2}}
E
(
f
(
x
)
)
,
E
(
g
(
x
)
)
{\displaystyle E(f(x)),E(g(x))}
E
(
f
(
x
)
⋅
g
(
y
)
)
=
E
(
f
(
x
)
)
⋅
E
(
(
y
)
)
{\displaystyle E(f(x)\cdot g(y))=E(f(x))\cdot E((y))}
Es gilt
N
(
μ
1
,
σ
1
2
)
×
N
(
μ
2
,
σ
2
2
)
=
N
(
μ
1
+
μ
2
,
σ
1
2
+
σ
2
2
)
{\displaystyle N(\mu _{1},\sigma _{1}^{2})\times N(\mu _{2},\sigma _{2}^{2})=N(\mu _{1}+\mu _{2},\sigma _{1}^{2}+\sigma _{2}^{2})}
Sei
X
1
{\displaystyle X_{1}}
N
(
μ
1
,
σ
1
2
)
{\displaystyle N(\mu _{1},\sigma _{1}^{2})}
X
2
{\displaystyle X_{2}}
N
(
μ
2
,
σ
2
)
{\displaystyle N(\mu _{2},\sigma ^{2})}
X
1
,
X
2
{\displaystyle X_{1},X_{2}}
ϕ
X
1
+
X
2
(
t
)
=
ϕ
X
1
⋅
ϕ
X
2
(
t
)
=
e
i
t
μ
1
⋅
e
1
2
σ
1
2
t
2
⋅
e
i
t
μ
2
⋅
e
−
1
2
σ
2
2
t
2
{\displaystyle \phi _{X_{1}+X_{2}}(t)=\phi _{X_{1}}\cdot \phi _{X_{2}}(t)=e^{it\mu _{1}}\cdot e^{{\frac {1}{2}}\sigma _{1}^{2}t^{2}}\cdot e^{it\mu _{2}}\cdot e^{-{\frac {1}{2}}\sigma _{2}^{2}t^{2}}}
=
e
i
t
(
μ
1
+
μ
2
)
⋅
e
−
1
2
(
σ
1
2
+
σ
2
2
)
t
2
{\displaystyle =e^{it(\mu _{1}+\mu _{2})}\cdot e^{-{\frac {1}{2}}(\sigma _{1}^{2}+\sigma _{2}^{2})t^{2}}}
⇒
Behauptung
.
{\displaystyle \Rightarrow {\text{Behauptung}}.}
Für die Zufallsvariable
X
{\displaystyle X}
E
(
X
m
)
{\displaystyle E(X^{m})}
∈
N
{\displaystyle \in \mathbb {N} }
ϕ
X
{\displaystyle \phi _{X}}
m
{\displaystyle m}
ϕ
X
(
m
)
(
0
)
=
i
m
E
(
X
m
)
{\displaystyle \phi _{X}^{(m)}(0)=i^{m}E(X^{m})}
(für
m
{\displaystyle m}
![{\displaystyle \Omega =[0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a16b39940ed262a459907035d290bf78c18134b8)
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\displaystyle P([a,b])=b-a,\quad 0\leq a\leq b\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/399b32ebd8ffb43c5d1289126bc98c3ffbbd4ba8)
![{\displaystyle P:{\mathcal {S}}([0,1])\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b04c32ff373b2fc4ca5b4db240fc11cf8ece62)
![{\displaystyle P:{\mathcal {S}}\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6f48da91c1db603d0d86088b3c858cffd5e454)
![{\displaystyle (a,b]=\otimes _{i=1}^{n}(a_{i},b_{i}]=\lbrace x\in \mathbb {R} ^{k}:a_{i}<x_{i}<b_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80e14a487c9ae5fbe901dce7d7512dcdd99cdc75)
![{\displaystyle {\mathcal {I}}^{k}=\lbrace (a,b],a<b\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7f99a90de609e0a98eb913b66fee3ab829b5dd)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle {\tilde {P}}:{\mathcal {I}}^{k}\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45ab2b7d58e624e3d2146f93c47ee47df866348c)
![{\displaystyle \Omega =[0,1]^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d89d54f460eb63167a3e8e2440cd863089742b7d)
![{\displaystyle \Omega =[0,2\pi ),\Omega \cap {\mathcal {I}}^{1}=\lbrace (a,b],0\leq a<b\leq 2\pi \rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc17b95b93d4ecb0c5e62fd527a8e26ea76e8cc8)
![{\displaystyle {\tilde {P}}(a,b]={\frac {b-a}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0492fe08756dca85686580da03eb2ea02ab512)
![{\displaystyle {\tilde {P}}(a,b]=e^{-\lambda a}e^{-\lambda b},0\leq a\leq b\leq \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a29b24e604da1ce6fe5dcc6a4ef5922b1b4d479)
![{\displaystyle (a,b]=(-\infty ,b]\setminus (-\infty ,a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5688a5eeab5f5390338c9ccb564497a0673c9b06)
![{\displaystyle P(a,b]=P(-\infty ,b]-P(-\infty ,a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11416139c1845a5dd743793b9e8b5dcaad8c9dc1)
![{\displaystyle F(t)=P(-\infty ,t],t\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7848b5b8c1fcd8d0174048d987c52feefd68fbef)
![{\displaystyle F_{P}:\mathbb {R} \to [0,1],F_{P}(t)=P(-\infty ,t],t\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84f995dbb292f4bbfee3a2dedaa09d08fc4ebe)
![{\displaystyle (t,t_{n}]=\bigcup _{i=n}^{\infty }(t_{i+1},t_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8c4c36780b675128e03bdfce276154737581c0)
![{\displaystyle F(t+)=lim_{n\to \infty }F(t_{n})=lim_{n\to \infty }[F(t)+P(t,t_{n}]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c8422477f931262cfde4f040340acfcc0de820)
![{\displaystyle =lim_{n\to \infty }[F(t)+\sum _{i=n}^{\infty }P(t_{i+1},t_{i}]]=F(t)+0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5da9e4261de7532dab64ab04002efd39b108d3d7)
![{\displaystyle \sum _{i=1}^{\infty }P(t_{i+1},t_{i}]=P(t,t_{1}]<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f926552b46b5cde49bf95799a4506231f6d1a25d)
![{\displaystyle (t_{1},t_{n}=\bigcup _{i=1}^{n-1}(t_{i},t_{i+1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745af2fdf037dcb875fc9a7d3e8a29c63eb03a5f)
![{\displaystyle F(t-)=lim_{n\to \infty }F(t_{n})=lim_{n\to \infty }[F(t_{1})+P(t_{1},t_{n}]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ba8cbee1ef1eef33a6fff598b826cb966f7ad7)
![{\displaystyle =lim_{n\to \infty }[F(t_{1}1)+\sum _{i=1}^{n-1}P(t_{i},t_{i+1}]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3606b33d29b28f223eb9c3fb2678a4df7b12bed7)
![{\displaystyle =F(t_{1})+P(\bigcup _{i=1}^{\infty }(t_{i},t_{i+1}])=F(t_{1})+P(t_{1},t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef13ba669a15c256a1ae10a6d12a38f4b9807610)
![{\displaystyle [a,b],(a,b),[a,b),(a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0678aee75cdfc79e850b0b03c6c8ca688b915f9a)
![{\displaystyle P(a,b]=F(b)-F(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d76fee68dedcc0f5ad6e939c8579dd0abe2bdf4d)
![{\displaystyle P[a,b]=P(b)-P(a-)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d25d31b9f3714673df5ba1479be44d7e9acd6956)
![{\displaystyle P[a,b]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9686a7df707fe8329d096951dc5e906d3bb96f22)
![{\displaystyle \Omega =[A,B]\subset \mathbb {R} ,A<B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c56b73479449cb7b330fe0db8832d88360a2db7b)
![{\displaystyle f(x)=\left\{{\begin{array}{ll}0,&x\notin [A,B]\\{\frac {1}{B-A}},&x\in [A,B]\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb12dc4f78faffe14d6950d14b95635990613da)
![{\displaystyle (a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6969e731af335df071e247ee7fb331cd1a57ae)
![{\displaystyle (a,b]=\otimes _{i=1}^{k}(a_{i},b_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9f16369c8b7ff6b7c2756c44557f96c9e4a6f4)
![{\displaystyle P(a,b]=\int _{(a,b]}f(x)d^{k}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e6f2df564b2574e5882773b35624bd76b6f0f5)
![{\displaystyle P:F^{k}\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f43dc23f14671269afe8ac9cbba7bf9737394e75)
![{\displaystyle P_{X}:{\mathcal {S}}'\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb08f6f20d7aa80eda4c23a8af2701314502a3f)
![{\displaystyle \lbrace X<b\rbrace \equiv X^{-1}(-\infty ,b]\in {\mathcal {S}},\forall b\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d15a8ccc638c40793174d268e2cd74c7f7ef304)
![{\displaystyle F'=\lbrace (-\infty ,b]:b\in \mathbb {R} \rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8ef8f833c634de267afe910fb971a4fa800a1c)
![{\displaystyle X_{i}^{-1}(a,b]=X^{-1}(\mathbb {R} \times ...\times \mathbb {R} \times (a,b]\times \mathbb {R} \times ...\times \mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33f57628374244169641b6b602a2505089a7a955)
![{\displaystyle F_{X}(x)=P(x\in X)=P_{X}(-\infty ,x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b64c4351ef26216c447a816a7484f488b5e76239)
![{\displaystyle P_{X}(a,b]=\int _{a_{k}}^{b_{k}}...\int _{a_{1}}^{b_{1}}f(x_{1},...,x_{k})dx_{1}...dx_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e37a36943c21c98b3085fb7cc8fed519cd19a5)
![{\displaystyle (a,b]=\otimes _{1}^{n}(a_{i},b_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91803e21e2bae49fa5d2a05a7202ddec2a6204cf)
![{\displaystyle [),[],()}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b86179eedbe926ebc16905196f4c92c36f155e)
![{\displaystyle P_{(x_{1},...,x_{n})}((a_{1},b_{1}]\times ...\times (a_{n},b_{n}])=P_{X_{1}}(a_{1},b_{1}]\cdot ...\cdot P_{X_{n}}(a_{n},b_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/315f59ebe1307e094ef91fd30efd5a26d78c78ab)
![{\displaystyle I_{k,n}=({\frac {k}{n}},{\frac {k+1}{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29760f81066bd00ca83d28c49c8ea1e48fb562f3)
![{\displaystyle [E(X\cdot Y)]^{2}\leq E(X)^{2}\cdot E(Y)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68474233405952c7cfd1e868d1d3325e69cc1b)
![{\displaystyle \phi _{X}(t)=(1-|t|)1_{[-1,1]}(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5c57d1bc92e6ebbd84313fa76383efe51801ad)
![{\displaystyle \phi (t)=1_{[-1,1]}(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3307eedd10a0b6b3f1a1d756bd42cfdc06d751c8)
![{\displaystyle g_{x}(s)=E(s^{x})=\sum _{k=k}^{\infty }sk\cdot P_{x}(\lbrace k\rbrace ),s\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b64d6ab43faf3995afe532dbed5ff88988b6d31)
![{\displaystyle =[(sin(tx)){\frac {1}{\sqrt {2\pi }}}(-x)e^{-{\frac {x^{2}}{2}}}]_{-\infty }^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15fdd8dc45a954bf56158ad52c58c42e746fb322)
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