Wir wollen zwei Ereignisse
A
,
B
{\displaystyle A,B}
P
(
B
|
A
)
{\displaystyle P(B|A)}
A
{\displaystyle A}
P
(
B
|
A
)
=
P
(
B
)
{\displaystyle P(B|A)=P(B)}
P
(
A
)
>
0
{\displaystyle P(A)>0}
Ist
P
{\displaystyle P}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
A
,
B
⊂
Ω
{\displaystyle A,B\subset \Omega }
A
{\displaystyle A}
B
{\displaystyle B}
P
(
A
∩
B
)
=
P
(
A
)
⋅
P
(
B
)
{\displaystyle P(A\cap B)=P(A)\cdot P(B)}
Ist
P
(
A
)
=
0
{\displaystyle P(A)=0}
B
{\displaystyle B}
A
,
B
{\displaystyle A,B}
P
(
A
∩
B
)
≤
P
(
A
)
{\displaystyle P(A\cap B)\leq P(A)}
Obige Gleichung ist im Fall
P
(
A
)
>
0
{\displaystyle P(A)>0}
P
(
B
|
A
)
=
P
(
B
)
{\displaystyle P(B|A)=P(B)}
Aus
A
,
B
{\displaystyle A,B}
(
A
¯
,
B
)
,
(
A
,
B
¯
)
,
(
A
¯
,
B
¯
)
{\displaystyle ({\bar {A}},B),(A,{\bar {B}}),({\bar {A}},{\bar {B}})}
Erweiterung der vorangegangenen Definition auf
n
{\displaystyle n}
P
{\displaystyle P}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
A
1
,
.
.
.
,
A
n
∈
S
{\displaystyle A_{1},...,A_{n}\in {\mathcal {S}}}
A
1
,
.
.
.
,
A
n
{\displaystyle A_{1},...,A_{n}}
J
:=
{
j
1
,
.
.
.
,
j
k
}
∩
{
1
,
.
.
.
,
n
}
{\displaystyle J:=\lbrace j_{1},...,j_{k}\rbrace \cap \lbrace 1,...,n\rbrace }
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})}
Ein Begriff, der in der Stochastik keine Rolle spielt, ist der der paarweisen Unabhängigkeit der
A
1
,
.
.
.
,
A
n
{\displaystyle A_{1},...,A_{n}}
P
(
A
i
∩
A
j
)
=
P
(
A
i
)
⋅
P
(
A
j
)
,
i
≠
j
{\displaystyle P(A_{i}\cap A_{j})=P(A_{i})\cdot P(A_{j}),i\neq j}
Daraus folgt nicht notwendigerweise die Unabhängigkeit der
A
1
,
.
.
.
,
A
n
{\displaystyle A_{1},...,A_{n}}
Folgende Eigenschaft der
A
1
,
.
.
.
A
n
{\displaystyle A_{1},...A_{n}}
{
j
1
,
.
.
.
,
j
k
}
⊂
{
1
,
.
.
.
,
n
}
{\displaystyle \lbrace j_{1},...,j_{k}\rbrace \subset \lbrace 1,...,n\rbrace }
P
(
A
j
1
∩
.
.
.
∩
A
j
k
)
>
0
{\displaystyle P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k})>0}
i
∈
{
1
,
.
.
.
,
n
}
∖
{
j
1
,
.
.
.
,
j
k
}
{\displaystyle i\in \lbrace 1,...,n\rbrace \setminus \lbrace j_{1},...,j_{k}\rbrace }
P
(
A
i
|
A
j
1
∩
.
.
.
∩
A
j
k
)
=
P
(
A
i
)
{\displaystyle P(A_{i}|{A_{j}}_{1}\cap ...\cap {A_{j}}_{k})=P(A_{i})}
Unabhängigkeitskriterium ist äquivalent zur stochastischen Unabhängigkeit der
A
1
,
.
.
.
,
A
n
{\displaystyle A_{1},...,A_{n}}
Aus der Definition folgt im Fall
P
(
A
j
1
∩
.
.
.
∩
A
j
k
)
>
0
{\displaystyle P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k})>0}
P
(
A
i
|
A
j
1
∩
.
.
.
∩
A
j
k
)
{\displaystyle P(A_{i}|{A_{j}}_{1}\cap ...\cap {A_{j}}_{k})}
=
P
(
A
j
1
∩
.
.
.
∩
A
j
k
∩
A
i
)
P
(
A
j
1
∩
.
.
.
∩
A
j
k
)
=
P
(
A
i
)
{\displaystyle ={\frac {P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k}\cap A_{i})}{P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k})}}=P(A_{i})}
Gilt umgekehrt
P
(
A
i
|
A
j
1
∩
.
.
.
∩
A
j
k
)
=
P
(
A
i
)
{\displaystyle P(A_{i}|{A_{j}}_{1}\cap ...\cap {A_{j}}_{k})=P(A_{i})}
P
(
A
j
1
∩
.
.
.
∩
A
j
k
−
1
)
=
P
(
A
i
)
>
0
{\displaystyle P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k-1})=P(A_{i})>0}
die Produktformel
P
(
A
j
1
∩
.
.
.
∩
A
j
k
)
=
P
(
A
j
1
)
⋅
P
(
A
j
2
|
A
j
1
)
⋅
P
(
A
j
k
|
A
j
1
∩
.
.
.
∩
A
j
k
−
1
)
{\displaystyle P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k})=P({A_{j}}_{1})\cdot P({A_{j}}_{2}|{A_{j}}_{1})\cdot P({A_{j}}_{k}|{A_{j}}_{1}\cap ...\cap {A_{j}}_{k-1})}
=
P
(
A
j
1
)
⋅
.
.
.
⋅
P
(
A
j
k
)
,
{\displaystyle =P({A_{j}}_{1})\cdot ...\cdot P({A_{j}}_{k}),}
Gilt
P
(
A
j
1
∩
.
.
.
∩
A
j
k
−
1
)
=
P
(
A
i
)
>
0
{\displaystyle P({A_{j}}_{1}\cap ...\cap {A_{j}}_{k-1})=P(A_{i})>0}
∅
{\displaystyle \varnothing }
Weißer und schwarzer Würfel. Gegeben
A
′
,
B
′
⊂
{
1
,
.
.
.
,
6
}
{\displaystyle A',B'\subset \lbrace 1,...,6\rbrace }
Ω
=
{
1
,
.
.
.
,
6
}
2
{\displaystyle \Omega =\lbrace 1,...,6\rbrace ^{2}}
A
=
A
′
×
{
1
,
.
.
.
,
6
}
{\displaystyle A=A'\times \lbrace 1,...,6\rbrace }
B
=
B
′
×
{
1
,
.
.
.
,
6
}
{\displaystyle B=B'\times \lbrace 1,...,6\rbrace }
A
′
{\displaystyle A'}
B
′
{\displaystyle B'}
P
{\displaystyle P}
Ω
{\displaystyle \Omega }
A
,
B
{\displaystyle A,B}
Erweiterung des Begriffs "Unabhängigkeit" für Ereignisse auf
Zufallsexperimente (Wahrscheinlichkeitsräume)
Zufallsgrößen Seien
(
Ω
1
,
S
1
,
P
1
)
{\displaystyle (\Omega _{1},{\mathcal {S_{1}}},P_{1})}
(
Ω
2
,
S
2
,
P
2
)
{\displaystyle (\Omega _{2},{\mathcal {S_{2}}},P_{2})}
Roulette ",
Ω
i
=
{
1
,
.
.
.
,
6
}
{\displaystyle \Omega _{i}=\lbrace 1,...,6\rbrace }
P
i
{\displaystyle P_{i}}
Ω
i
{\displaystyle \Omega _{i}}
(
i
=
1
,
2
)
{\displaystyle (i=1,2)}
(
Ω
,
P
)
{\displaystyle (\Omega ,P)}
Ω
=
{
1
,
.
.
.
,
6
}
×
{
0
,
1
,
.
.
.
,
36
}
{\displaystyle \Omega =\lbrace 1,...,6\rbrace \times \{0,1,...,36\}}
P
{\displaystyle P}
Ω
{\displaystyle \Omega }
P
(
A
1
×
A
2
)
=
P
1
(
A
1
)
⋅
P
2
(
A
2
)
{\displaystyle P(A_{1}\times A_{2})=P_{1}(A_{1})\cdot P_{2}(A_{2})}
P
:
S
→
[
0
,
1
]
{\displaystyle P:{\mathcal {S}}\to [0,1]}
S
:=
℘
(
Ω
1
×
Ω
2
)
{\displaystyle {\mathcal {S}}:=\wp (\Omega _{1}\times \Omega _{2})}
für alle
A
1
⊂
Ω
1
{\displaystyle A_{1}\subset \Omega _{1}}
A
2
⊂
Ω
2
{\displaystyle A_{2}\subset \Omega _{2}}
Dies gibt Anlass zur folgenden Definition.
Bearbeiten
Gegeben sind (diskrete) Wahrscheinlichkeitsräume
(
Ω
1
,
S
1
,
P
1
)
{\displaystyle (\Omega _{1},{\mathcal {S_{1}}},P_{1})}
(
Ω
n
,
S
n
,
P
n
)
{\displaystyle (\Omega _{n},{\mathcal {S_{n}}},P_{n})}
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
Ω
=
Ω
1
×
.
.
.
×
Ω
n
{\displaystyle \Omega =\Omega _{1}\times ...\times \Omega _{n}}
P
{\displaystyle P}
S
{\displaystyle {\mathcal {S}}}
P
(
A
1
×
.
.
.
×
A
n
)
=
P
1
(
A
1
)
⋅
.
.
.
⋅
P
n
(
A
n
)
{\displaystyle P(A_{1}\times ...\times A_{n})=P_{1}(A_{1})\cdot ...\cdot P_{n}(A_{n})}
für alle
A
1
∈
S
1
,
.
.
.
,
A
n
∈
S
n
{\displaystyle A_{1}\in {\mathcal {S}}_{1},...,A_{n}\in {\mathcal {S}}_{n}}
(
Ω
i
,
S
i
,
P
i
)
{\displaystyle (\Omega _{i},{\mathcal {S_{i}}},P_{i})}
i
=
1
,
.
.
.
,
n
{\displaystyle i=1,...,n}
S
{\displaystyle {\mathcal {S}}}
A
1
×
.
.
.
×
A
n
{\displaystyle A_{1}\times ...\times A_{n}}
A
1
∈
S
1
,
{\displaystyle A_{1}\in {\mathcal {S}}_{1},}
A
n
∈
S
n
{\displaystyle A_{n}\in {\mathcal {S}}_{n}}
Sind
(
Ω
1
,
S
1
,
P
1
)
,
.
.
.
,
(
Ω
n
,
S
n
,
P
n
)
{\displaystyle (\Omega _{1},{\mathcal {S}}_{1},P_{1}),...,(\Omega _{n},{\mathcal {S}}_{n},P_{n})}
P
{\displaystyle P}
Ω
=
Ω
1
×
.
.
.
×
Ω
n
{\displaystyle \Omega =\Omega _{1}\times ...\times \Omega _{n}}
S
{\displaystyle {\mathcal {S}}}
Jede diskrete Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
(
Ω
,
S
)
{\displaystyle (\Omega ,{\mathcal {S}})}
q
(
w
)
,
w
=
(
w
1
,
.
.
.
,
w
n
)
∈
Ω
{\displaystyle q(w),w=(w_{1},...,w_{n})\in \Omega }
q
(
w
)
=
P
(
{
w
}
)
=
P
(
{
w
1
}
×
.
.
.
×
{
w
n
}
)
=
P
1
(
{
w
1
}
)
⋅
.
.
.
⋅
P
(
{
w
n
}
)
{\displaystyle q(w)=P(\lbrace w\rbrace )=P(\lbrace w_{1}\rbrace \times ...\times \lbrace w_{n}\rbrace )=P_{1}(\lbrace w_{1}\rbrace )\cdot ...\cdot P(\lbrace w_{n}\rbrace )}
Man definiere die Abbildung
q
:
Ω
→
[
0
,
1
]
{\displaystyle q:\Omega \to [0,1]}
q
(
w
)
=
P
(
{
w
}
)
=
P
1
(
{
w
1
}
)
⋅
.
.
.
⋅
P
(
{
w
n
}
)
{\displaystyle q(w)=P(\lbrace w\rbrace )=P_{1}(\lbrace w_{1}\rbrace )\cdot ...\cdot P(\lbrace w_{n}\rbrace )}
q
{\displaystyle q}
Ω
{\displaystyle \Omega }
P
{\displaystyle P}
Ω
{\displaystyle \Omega }
P
{\displaystyle P}
P
(
A
1
×
.
.
.
×
A
n
)
=
∑
w
1
∈
A
1
,
.
.
.
,
w
n
∈
A
n
q
(
(
w
1
,
.
.
.
,
w
n
)
)
{\displaystyle P(A_{1}\times ...\times A_{n})=\sum _{w_{1}\in A_{1},...,w_{n}\in A_{n}}q((w_{1},...,w_{n}))}
=
∑
w
1
∈
A
1
P
1
(
{
w
1
}
)
⋅
.
.
.
⋅
∑
w
n
∈
A
n
P
(
{
w
n
}
)
{\displaystyle =\sum _{w_{1}\in A_{1}}P_{1}(\lbrace w_{1}\rbrace )\cdot ...\cdot \sum _{w_{n}\in A_{n}}P(\lbrace w_{n}\rbrace )}
=
P
1
(
A
1
)
⋅
.
.
.
⋅
P
n
(
A
n
)
.
{\displaystyle =P_{1}(A_{1})\cdot ...\cdot P_{n}(A_{n}).}
Die Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
Ω
1
×
.
.
.
×
Ω
n
=
Ω
{\displaystyle \Omega _{1}\times ...\times \Omega _{n}=\Omega }
P
=
P
1
×
.
.
.
×
P
n
=
⊗
i
=
1
n
P
i
{\displaystyle P=P_{1}\times ...\times P_{n}=\otimes _{i=1}^{n}P_{i}}
P
1
,
.
.
.
,
P
n
{\displaystyle P_{1},...,P_{n}}
1. Die einzelnen Faktoren
P
1
,
.
.
.
,
P
n
{\displaystyle P_{1},...,P_{n}}
P
=
⊗
i
=
1
n
P
i
{\displaystyle P=\otimes _{i=1}^{n}P_{i}}
Randverteilungen genannt) lassen sich (wie folgt, aus
P
{\displaystyle P}
Ist nämlich
A
i
∈
Ω
{\displaystyle A_{i}\in \Omega }
i
=
{
1
,
.
.
.
,
n
}
{\displaystyle i=\lbrace 1,...,n\rbrace }
A
=
Ω
1
×
.
.
.
×
Ω
i
−
1
×
A
i
×
Ω
i
+
1
×
.
.
.
×
Ω
n
{\displaystyle A=\Omega _{1}\times ...\times \Omega _{i-1}\times A_{i}\times \Omega _{i+1}\times ...\times \Omega _{n}}
P
(
A
)
=
1
⋅
.
.
.
⋅
1
⋅
P
(
A
i
)
⋅
.
.
.
⋅
1
⋅
1
=
P
i
(
A
i
)
{\displaystyle P(A)=1\cdot ...\cdot 1\cdot P(A_{i})\cdot ...\cdot 1\cdot 1=P_{i}(A_{i})}
2. Spezialfall:
Ω
1
,
.
.
.
,
Ω
n
{\displaystyle \Omega _{1},...,\Omega _{n}}
P
=
⊗
i
=
1
n
P
i
{\displaystyle P=\otimes _{i=1}^{n}P_{i}}
Ω
=
Ω
1
×
.
.
.
×
Ω
n
{\displaystyle \Omega =\Omega _{1}\times ...\times \Omega _{n}}
P
i
,
(
i
=
1
,
.
.
.
,
n
)
{\displaystyle P_{i},(i=1,...,n)}
Gleichverteilung auf
Ω
i
{\displaystyle \Omega _{i}}
In der Tat:
P
(
{
w
}
)
=
1
|
Ω
1
|
⋅
.
.
.
⋅
|
Ω
n
|
{\displaystyle P(\lbrace w\rbrace )={\frac {1}{|\Omega _{1}|\cdot ...\cdot |\Omega _{n}|}}}
w
=
(
w
1
,
.
.
.
,
w
n
)
∈
Ω
{\displaystyle w=(w_{1},...,w_{n})\in \Omega }
(Definition Produkt WKT-Räume)
⇑
{\displaystyle \Uparrow }
⇓
{\displaystyle \Downarrow }
P
i
(
{
w
i
}
)
=
1
|
Ω
|
{\displaystyle P_{i}(\lbrace w_{i}\rbrace )={\frac {1}{|\Omega |}}}
i
=
1
,
.
.
.
,
n
,
w
i
∈
Ω
{\displaystyle i=1,...,n,w_{i}\in \Omega }
3. Ist
Ω
1
=
.
.
.
=
Ω
n
,
P
1
=
.
.
.
=
P
n
{\displaystyle \Omega _{1}=...=\Omega _{n},P_{1}=...=P_{n}}
Ω
1
n
,
⊗
i
=
1
n
P
i
{\displaystyle \Omega _{1}^{n},\otimes _{i=1}{n}P_{i}}
Ω
1
,
P
1
{\displaystyle \Omega _{1},P_{1}}
4. Es gibt viele Möglichkeiten, eine Wahrscheinlichkeitsverteilung
P
{\displaystyle P}
Ω
1
×
.
.
.
×
Ω
n
{\displaystyle \Omega _{1}\times ...\times \Omega _{n}}
⊗
i
=
1
n
P
i
{\displaystyle \otimes _{i=1}^{n}P_{i}}
Ω
1
=
.
.
.
=
Ω
n
,
P
1
=
.
.
.
=
P
n
{\displaystyle \Omega _{1}=...=\Omega _{n},P_{1}=...=P_{n}}
P
{\displaystyle P}
Ω
1
n
{\displaystyle \Omega _{1}^{n}}
P
(
(
w
1
,
.
.
.
,
w
n
)
)
=
{
P
1
(
{
w
1
}
)
,
w
1
=
.
.
.
=
w
n
0
,
s
o
n
s
t
.
{\displaystyle P((w_{1},...,w_{n}))=\left\{{\begin{array}{ll}P_{1}(\lbrace w_{1}\rbrace ),&w_{1}=...=w_{n}\\0,&sonst\end{array}}\right..}
P
{\displaystyle P}
P
{\displaystyle P}
w
1
{\displaystyle w_{1}}
0
<
P
1
(
{
w
1
}
)
<
1
{\displaystyle 0<P_{1}(\lbrace w_{1}\rbrace )<1}
n
{\displaystyle n}
w
1
{\displaystyle w_{1}}
P
(
{
w
1
}
×
.
.
.
×
{
w
1
}
)
{\displaystyle P(\lbrace w_{1}\rbrace \times ...\times \lbrace w_{1}\rbrace )}
=
P
(
{
(
w
1
,
.
.
.
,
w
1
)
}
)
=
P
1
(
{
w
1
}
)
≠
[
P
1
(
{
w
1
}
)
]
n
.
{\displaystyle =P(\lbrace (w_{1},...,w_{1})\rbrace )=P_{1}(\lbrace w_{1}\rbrace )\neq [P_{1}(\lbrace w_{1}\rbrace )]^{n}.}
Der Wahrscheinlichkeitsraum
(
Ω
1
n
,
P
)
{\displaystyle (\Omega _{1}^{n},P)}
P
{\displaystyle P}
n
{\displaystyle n}
n
{\displaystyle n}
Auf dem Wahrscheinlichkeitsraum
(
Ω
,
S
,
P
)
{\displaystyle (\Omega ,{\mathcal {S}},P)}
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
X
i
:
Ω
→
Ω
′
,
i
=
1
,
.
.
.
,
n
{\displaystyle X_{i}:\Omega \to \Omega ',i=1,...,n}
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
P
(
X
1
−
1
(
B
1
)
∩
.
.
.
∩
X
n
−
1
(
B
n
)
)
=
P
(
X
1
−
1
(
B
1
)
)
⋅
.
.
.
⋅
P
(
X
n
−
1
(
B
n
)
)
{\displaystyle P(X_{1}^{-1}(B_{1})\cap ...\cap X_{n}^{-1}(B_{n}))=P(X_{1}^{-1}(B_{1}))\cdot ...\cdot P(X_{n}^{-1}(B_{n}))}
für alle
B
i
⊂
Ω
i
,
i
=
1
,
.
.
.
,
n
{\displaystyle B_{i}\subset \Omega _{i},i=1,...,n}
1. Für unabhängige
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
X
1
−
1
(
B
1
)
,
.
.
.
,
X
n
−
1
(
B
n
)
{\displaystyle X_{1}^{-1}(B_{1}),...,X_{n}^{-1}(B_{n})}
J
=
{
j
1
,
.
.
.
,
j
k
}
⊂
{
1
,
.
.
.
,
n
}
{\displaystyle J=\lbrace j_{1},...,j_{k}\rbrace \subset \lbrace 1,...,n\rbrace }
B
j
=
Ω
j
′
{\displaystyle B_{j}=\Omega '_{j}}
j
∉
J
{\displaystyle j\notin J}
X
j
−
1
(
Ω
j
′
)
=
Ω
{\displaystyle X_{j}^{-1}(\Omega '_{j})=\Omega }
P
(
X
j
1
−
1
(
B
j
1
)
∩
.
.
.
∩
X
j
k
−
1
(
B
j
k
)
)
=
P
(
X
j
1
−
1
(
B
j
1
)
)
⋅
.
.
.
⋅
P
(
X
j
k
−
1
(
B
j
k
)
)
{\displaystyle P({X_{j}}_{1}^{-1}({B_{j}}_{1})\cap ...\cap {X_{j}}_{k}^{-1}({B_{j}}_{k}))=P({X_{j}}_{1}^{-1}({B_{j}}_{1}))\cdot ...\cdot P({X_{j}}_{k}^{-1}({B_{j}}_{k}))}
2. Die Definition lässt sich auch auf folgende Weisen schreiben:
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})\in B_{1}\times ...\times B_{n})=P(X_{1}\in B_{1})\cdot ...\cdot P(X_{n}\in B_{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}(B_{1})\cdot ...\cdot {P_{X}}_{n}(B_{n})}
B
i
⊂
Ω
i
′
,
i
=
1
,
.
.
.
,
n
{\displaystyle B_{i}\subset \Omega '_{i},i=1,...,n}
Die letzte Gleichung führt unter Berücksichtigung der Definition der Produktverteilung zu folgendem Satz.
Die Zufallsgrößen
X
1
,
.
.
.
,
X
n
,
X
i
:
Ω
→
Ω
i
′
,
i
=
1
,
.
.
.
,
n
{\displaystyle X_{1},...,X_{n},X_{i}:\Omega \to \Omega '_{i},i=1,...,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}.}
(In Worten: Falls die gemeinsame Verteilung der
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
Sind
X
1
,
.
.
.
,
X
n
{\displaystyle X_{1},...,X_{n}}
ϕ
1
,
.
.
.
,
ϕ
n
,
ϕ
i
:
R
→
R
,
i
=
1
,
.
.
.
,
n
{\displaystyle \phi _{1},...,\phi _{n},\phi _{i}:\mathbb {R} \to \mathbb {R} ,i=1,...,n}
ϕ
1
∘
X
1
,
.
.
.
,
ϕ
n
∘
X
n
{\displaystyle \phi _{1}\circ X_{1},...,\phi _{n}\circ X_{n}}
Wegen
(
ϕ
i
∘
X
i
)
−
1
(
B
i
)
=
X
i
−
1
(
ϕ
−
1
(
B
i
)
)
,
B
i
⊂
R
{\displaystyle (\phi _{i}\circ X_{i})^{-1}(B_{i})=X_{i}^{-1}(\phi ^{-1}(B_{i})),B_{i}\subset \mathbb {R} }
P
(
(
ϕ
1
∘
X
1
)
−
1
(
B
1
)
∩
.
.
.
∩
(
ϕ
n
∘
X
n
)
−
1
(
B
n
)
)
{\displaystyle P((\phi _{1}\circ X_{1})^{-1}(B_{1})\cap ...\cap (\phi _{n}\circ X_{n})^{-1}(B_{n}))}
=
P
(
X
1
−
1
(
ϕ
1
−
1
(
B
1
)
)
∩
.
.
.
∩
X
n
−
1
(
ϕ
n
−
1
(
B
n
)
)
)
{\displaystyle =P(X_{1}^{-1}(\phi _{1}^{-1}(B_{1}))\cap ...\cap X_{n}^{-1}(\phi _{n}^{-1}(B_{n})))}
=
P
(
X
1
−
1
(
ϕ
1
−
1
(
B
1
)
)
)
⋅
.
.
.
⋅
P
(
X
n
−
1
(
ϕ
n
−
1
(
B
n
)
)
)
{\displaystyle =P(X_{1}^{-1}(\phi _{1}^{-1}(B_{1})))\cdot ...\cdot P(X_{n}^{-1}(\phi _{n}^{-1}(B_{n})))}
=
P
(
(
ϕ
1
∘
X
1
)
−
1
)
⋅
.
.
.
⋅
P
(
(
ϕ
n
∘
X
n
)
−
1
(
B
n
)
)
.
{\displaystyle =P((\phi _{1}\circ X_{1})^{-1})\cdot ...\cdot P((\phi _{n}\circ X_{n})^{-1}(B_{n})).}
Sind also die beiden Zufallsvariablen
X
,
Y
{\displaystyle X,Y}
|
X
|
,
Y
2
{\displaystyle |X|,Y^{2}}
e
X
,
s
i
n
(
Y
)
{\displaystyle e^{X},sin(Y)}
X
+
Y
,
X
−
Y
{\displaystyle X+Y,X-Y}
![{\displaystyle P:{\mathcal {S}}\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6f48da91c1db603d0d86088b3c858cffd5e454)
![{\displaystyle q:\Omega \to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7a5c884d6c6283159dd2ba7874b0237f472b2a)
![{\displaystyle =P(\lbrace (w_{1},...,w_{1})\rbrace )=P_{1}(\lbrace w_{1}\rbrace )\neq [P_{1}(\lbrace w_{1}\rbrace )]^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c3c925db76b94f392d6f3a5da4c0a61b775bca)
