From 35656c065c071023fb061d028fbc80cc78115f22 Mon Sep 17 00:00:00 2001 From: Ameyah Date: Wed, 24 Apr 2019 17:15:19 +0200 Subject: [PATCH] lecture STOCH 24 April 2019 added --- 3. Semester/GEO/Vorlesung GEO.pdf | Bin 174837 -> 1030256 bytes .../Bedingte_Wahrscheinlichkeiten.tex | 2 +- .../STOCH/TeX_files/Erste_Standardmodelle.tex | 1 - .../TeX_files/Grundbegriffe_Wtheorie.tex | 2 +- .../unbedingte_Wahrscheinlichkeiten.tex | 143 ++++++++++++++++-- 5 files changed, 134 insertions(+), 14 deletions(-) diff --git a/3. Semester/GEO/Vorlesung GEO.pdf b/3. 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Semester/STOCH/TeX_files/Bedingte_Wahrscheinlichkeiten.tex b/4. Semester/STOCH/TeX_files/Bedingte_Wahrscheinlichkeiten.tex index 2d9e47f..bf29a4f 100644 --- a/4. Semester/STOCH/TeX_files/Bedingte_Wahrscheinlichkeiten.tex +++ b/4. Semester/STOCH/TeX_files/Bedingte_Wahrscheinlichkeiten.tex @@ -1,4 +1,4 @@ -\part{title}\section{Bedingte Wahrscheinlichkeiten} +\section{Bedingte Wahrscheinlichkeiten} \begin{example} \proplbl{3_1_1} Das Würfeln mit zwei fairen, sechsseitigen Würfeln können wir mit diff --git a/4. Semester/STOCH/TeX_files/Erste_Standardmodelle.tex b/4. Semester/STOCH/TeX_files/Erste_Standardmodelle.tex index ebf0ce9..306af81 100644 --- a/4. Semester/STOCH/TeX_files/Erste_Standardmodelle.tex +++ b/4. Semester/STOCH/TeX_files/Erste_Standardmodelle.tex @@ -1,5 +1,4 @@ \chapter{Erste Standardmodelle der Wahrscheinlichkeitstheorie} - %\subsection{Diskrete Verteilungen} \section*{Diskrete Verteilungen} diff --git a/4. Semester/STOCH/TeX_files/Grundbegriffe_Wtheorie.tex b/4. Semester/STOCH/TeX_files/Grundbegriffe_Wtheorie.tex index 7ccc136..f89bc4e 100644 --- a/4. Semester/STOCH/TeX_files/Grundbegriffe_Wtheorie.tex +++ b/4. Semester/STOCH/TeX_files/Grundbegriffe_Wtheorie.tex @@ -1,5 +1,5 @@ \chapter{Grundbegriffe der Wahrscheinlichkeitstheorie} - +\label{chapter1} \section{Wahrscheinlichkeitsräume} \subsection*{Ergebnisraum} diff --git a/4. Semester/STOCH/TeX_files/unbedingte_Wahrscheinlichkeiten.tex b/4. Semester/STOCH/TeX_files/unbedingte_Wahrscheinlichkeiten.tex index dbc6e2c..3c8f639 100644 --- a/4. Semester/STOCH/TeX_files/unbedingte_Wahrscheinlichkeiten.tex +++ b/4. Semester/STOCH/TeX_files/unbedingte_Wahrscheinlichkeiten.tex @@ -124,9 +124,7 @@ In vielen Fällen besagt die Intuition über verschiedene Zufallsexperimente / E \end{align*} \end{itemize} \end{proof} - Insbesondere zeigt \propref{3_2_16}, dass wir in einer Familie unabhängiger Ereignisse beliebig viele Ereignisse durch ihr Komplement, $\emptyset$ oder $\O$ ersetzen können, ohne die Unabhängigkeit zu verlieren. - \begin{proposition} \proplbl{3_2_17} Sei $(\O, \F, \P)$ Wahrscheinlichkeitsraum und $\F_i \subseteq \F, i \in I$, seien $\cap$-stabile Familien von Ereignissen. Dann gilt @@ -134,7 +132,6 @@ Insbesondere zeigt \propref{3_2_16}, dass wir in einer Familie unabhängiger Ere \F_i, i \in I \text{ unabhängig } \Leftrightarrow \sigma(\F_i), i \in I \text{ unabhängig}. \end{align*} \end{proposition} - \begin{proof} oBdA sei $I = \set{1, \dots, n}$ und $\O \in \F_i, i \in I$. \begin{itemize} @@ -147,23 +144,22 @@ Insbesondere zeigt \propref{3_2_16}, dass wir in einer Familie unabhängiger Ere \end{align*} \item Da die Familien $\F_i$ unabhängig sind, gilt $\mu\mid_{\F_1} = \nu\mid_{\F_1}$. - Nach dem Eindeutigkeitssatz für Maße (\proplbl{1_1_19}) folgt $\mu\mid_{\sigma(\F_1)} = \nu\mid_{\sigma(\F_1)}$ also + Nach dem Eindeutigkeitssatz für Maße (\propref{1_1_19}) folgt $\mu\mid_{\sigma(\F_1)} = \nu\mid_{\sigma(\F_1)}$ also \begin{align*} \P\brackets{ F \cap F_2 \cap \cdots \cap F_n} = \P(F) \P(F_2) \dots \P(F_n) \end{align*} - für alle $F \in \sigma(\F_i)$ und $F_i \in \F_i, i = 1, \dots, n$. Da $\O \in \F_i$ für alle $i$ gilt die erhaltene Produktformel auf für alle Teilemengen $J \subseteq I$.\\ + für alle $F \in \sigma(\F_i)$ und $F_i \in \F_i, i = 1, \dots, n$. Da $\O \in \F_i$ für alle $i$ gilt die erhaltene Produktformel für alle Teilemengen $J \subseteq I$.\\ Also sind \begin{align*} - \sigma(\F_1), \F_2, \dots, \F_n \text{unabhängig} + \sigma(\F_1), \F_2, \dots, \F_n \text{ unabhängig} \end{align*} \item Wiederholtes Anwenden von $1$ und $2$ liefert den Satz. \end{enumerate} \end{itemize} \end{proof} - Mit \propref{3_2_17} folgen: - \begin{conclusion} + \proplbl{3_2_18} Sei $(\O,\F,\P)$ Wahrscheinlichkeitsraum und \begin{align*} \F_{i,j} \subseteq \F, \quad 1 \le i \le n, 1 \le j \le m(i) @@ -175,8 +171,8 @@ Mit \propref{3_2_17} folgen: \end{align*} unabhängig. \end{conclusion} - \begin{conclusion} + \proplbl{3_2_19} Sei $(\O,\F,\P)$ Wahrscheinlichkeitsraum und \begin{align*} X_{ij}: \O \to E, \quad 1 \le i \le n, 1 \le j \le m(i) @@ -187,11 +183,136 @@ Mit \propref{3_2_17} folgen: \end{align*} unabhängig. \end{conclusion} - \begin{example} $X_1, \dots, X_n$ unabhängige reelle Zufallsvariablen. Dann sind auch \begin{align*} Y_1 = X_1, Y_2 = X_2 + \cdots + X_n \end{align*} unabhängig. -\end{example} \ No newline at end of file +\end{example} +% % % % % % % % % % % % % % % % % 7th lecture % % % % % % % % % % % % % % % % % % % +\begin{proof}[\propref{3_2_18}] + OBdA sei $\Omega \in \F_{i,j} \forall i,j$. Dann sind die Familien: + \begin{align*} + \F_i^{\cap} := \set{F_{i,1} \cap \dots \cap F_{i,m(i)} \mid F_{i,j} \in \F_{i,j}, 1 \le j \le m(i)}, 1 \le i \le n + \end{align*} + $\cap$-stabil, unabhängig und es gilt: $\F_{i,1}, \dots, \F_{i,m(i)} \subseteq \F_i^{\cap}$ ($\nearrow$ HA)! Nach \propref{3_2_17} sind auch $\sigma(\F_i^{\cap})$ unabhängig. Damit folgt die Behauptung, da $\sigma(\F_i^{\cap}) = \G_i$: + \begin{align*} + \F_{i,1}, \dots, \F_{i,m(i)} \subseteq \F_i^{\cap} \subseteq \sigma(\F_{i,1}, \dots, \F_{i,m(i)}) = \G_i\\ + \Rightarrow \G = \sigma(\F_{i,1}, \dots, \F_{i,m(i)}) \subseteq \sigma(\F_i^{\cap}) \subseteq \G_i. + \end{align*} +\end{proof} +\begin{proof}[\propref{3_2_19}] + Setze $\F_{i,j} = \sigma(X_{i,j})$ und $\G_i = \sigma(\F_{i,1}, \dots, \F_{i,m(i)})$, dann sind nach \propref{3_2_18} die $\G_i, i = 1,\dots,n$ unabhängig. Zudem ist + \begin{align*} + Y_i := f_i(X_{i,1}, \dots, X_{i,m(i)}) + \end{align*} + $\G_i$ messbar, also $\sigma(Y_i) \subseteq \G_i$. Damit erben die $Y_i$ die Unabhängigkeit der $\G_i$. +\end{proof} +\begin{proposition}[Unabhängigkeit von Zufallsvariablen] + \proplbl{3_2_20} + $(\O, \F, \P)$ Wahrscheinlichkeitsraum und $X_1, \dots, X_n: (\O,\F) \to (E, \E)$ Zufallsvariablen. Dann sind + \begin{enumerate} + \item $X_1, \dots, X_n$ sind unabhängig \label{prop:unabhZV1} + \item $\P(X_1 \in A_1, \dots, X_n \in A_n) = \prod_{i=1}^n \P(X_i \in A_i)\quad \forall A_1, \dots, A_n \in \E$. \label{prop:unabhZV2} + \item Die gemeinsame Verteilung der $X_i$ entspricht den Produktmaß der einzelnen Verteilungen \label{prop:unabhZV3} + \begin{align*} + \P_{X_1, \dots, X_n} = \bigotimes_{i=1}^n \P_{X_i} + \end{align*} + \end{enumerate} +\end{proposition} +\begin{proof} + Per Ringschluss: + \begin{enumerate} %TODO add labels + \item[1 $\Rightarrow$ 2:] Seien $A_1, \dots, A_n\in E$ beliebig, dann gilt per Definition + \begin{align*} % für Rechtecke + \P_{X_1, \dots, X_n}(A_1 \times \dots \times A_n) &= \P(X_1 \in A_1, \dots, X_n \in A_n)\\ + &= \P\brackets{\bigcap_{i=1}^n \set{X_i \in A_i}}\\ + &\overset{\text{unabh}}{=} \prod_{i=1}^n \P(X_i \in A_i)\\ + &= \prod_{i=1}^n \P_{X_i}(A_i) = \brackets{\bigotimes_{i=1}^n \P_{X_i}} (A_1 \times \dots \times A_n) + \end{align*} + \item[2 $\Rightarrow$ 3:] Aus der obigen Rechnung sehen wir, dass 2 bereits 3 impliziert für alle Rechtecke: $\bigtimes_{i = 1}^n A_i$. Da die Familie der Rechtecke $\cap$-stabil ist und $\E^{\otimes n}$ erzeugt, folgt die Aussage aus dem Eindeutigkeitssatz für Maße \propref{1_1_19}. + \item[3 $\Rightarrow$ 1:] Sei $J \subseteq \set{1,\dots,n}$ und setze + \begin{align*} + A_i &:= \begin{cases} + \text{ beliebig } &\in \E, i \in J\\ + E & i \notin J. + \end{cases} + \intertext{Dann} + \P(X_i \in A_i, i \in J) &= \P(X_i \in A_i, i = 1, \dots,n)\\ + &= \prod_{i=1}^n \P(X_i \in A_i)\\ + &= \prod_{i\in J} \P(A_i \in A_i). + \end{align*} + \end{enumerate} +\end{proof} +\begin{example} + Im Urnenmodell mit Zurücklegen hat der Vektor $X = (X_1, \dots, X_n)$ mit $X_i = $ Farbe im $i$-ten Zug als Zähldichte die Produktdichte der $X_i$. Die $X_1, \dots, X_n$ sind also unabhängig. +\end{example} +\subsection*{Konstruktion unabhängiger Zufallsvariablen} +Kapitel \ref{chapter1}: Zu beliebiger Wahrscheinlichkeitsverteilung $\P_X$ existiert Wahrscheinlichkeitsraum mit Zufallsvariable $X$ auf diesem Wahrscheinlichkeitsraum, so dass $X \sim \P_X$. +\begin{enumerate} + \item Seien $\P{X_1}, \dots, \P_{X_n}$ Wahrscheinlichkeitsverteilungen auf $(E, \E)$. Gibt es einen Wahrscheinlichkeitsraum $(\O, \F, \P)$ und Zufallsvariablen $X_1, X_2$ unabhängig, so dass $X_1 \sim \P_{X_1}$? \label{konstruktionunabh:ZV:qu_1} + \item Wie kann ich beliebig (unendlich) viele unabhängige Zufallsvariablen konstruieren? \label{konstruktionunabh:ZV:qu_2} +\end{enumerate} +Wir beginnen mit \ref{konstruktionunabh:ZV:qu_1}:\\ +Konstruiere zwei Wahrscheinlichkeitsräume $(\O_i, \F_i, \P_i), i = 1,2$ und Zufallsvariablen $X_1, X_2 \mit X_i \sim \P_{X_i}$. Auf dem Produktraum +\begin{align*} + \O = \O_1 \times \O_2, \quad \F := \F_1 \otimes \F_2 \und \P = \P_1 \otimes \P_2 + \intertext{ definiere} + X'_1: \O_1 \times \O_2 \to E\colon (\omega_1, \omega_2) \mapsto X_1(\omega_1)\\ + X'_2: \O_1 \times \O_2 \to E\colon (\omega_1, \omega_2) \mapsto X_2(\omega_2) +\end{align*} +Dann gilt für beliebige Ereignisse: $F_1, F_2 \in \E$ +\begin{align*} + \underbrace{\set{X'_1 \in F_1} \cap \set{X'_2 \in F_2}}_{\supseteq \O = \O_1 \times \O_2} = \underbrace{\set{X_1 \in F_1}}_{\supseteq \O_1} \times \underbrace{\set{X_2 \in F_2}}_{\supseteq \O_2} \in \F_1 \times \F_2 +\end{align*} +und damit folgt die Messbarkeit der Abbildungen $X'_1, X'_2$, d.h. $X'_1, X'_2$ sind Zufallsvariablen auf $(\O, \F)$. Zudem gilt +\begin{align*} + \P(X'_1 \in F_1, X'_2 \in F_2) &= \P_1 \otimes \P_2 \brackets{\set{X_1 \in F_1} \times \set{X_2 \in F_2}}\\ + &= \P_1 (X_1 \in F_1) \P_2(X_2 \in F_2), + \intertext{also} + \P(X'_i \in F_i) &= \P_i (X'_i \in F_i) +\end{align*} +sowie nach \propref{prop:Kolmo} $X'_1 \upmodels X'_2$.\\ +Wenn $(\O_2, \F_1, \P_1) = (\O_2, \F_2, \P_2)$, so liefert die obige Konstruktion zwei unabhängige Zufallsvariablen auf einem Wahrscheinlichkeitsraum. Andernfalls können wir auf den Produktraum ausweichen und $X'_i$ anstelle von $X_i$ betrachten. Die obige Konstruktion lässt sich direkt auf \emph{endlich} viele Zufallsvariablen übertragen.\\ +Zu \ref{konstruktionunabh:ZV:qu_2}: +\begin{proposition}[Satz von \person{Kolmogorov}] + \proplbl{prop:Kolmo} + Sei $I$ beliebige Indexmenge und $(\O_i, \F_i, \P_i), i \in I$ Wahrscheinlichkeitsräume. Setze + \begin{align*} + \O_I &:= \bigtimes_{i \in I} \O_i = \set{\omega : I \to \bigcup_{i \in I} \O_i, \omega_i \in \O_i, i \in I}\\ + \F_I &:= \sigma( \pi^{-1} ( \F_i), i \in I) + \end{align*} + wobei $\pi_i : \O_I \to \O_i \mit \omega \mapsto \omega_i$ die Projektionsabbildung. Dann existiert auf $(\O_I, \F_I)$ genau ein Maß $\P_I$, sodass für alle $H \subseteq I$ mit $0 < \abs{H} < \infty$ gilt + \begin{align*} + \pi_H ( \P_I) = \bigotimes_{i \in H} \P_i, + \end{align*} + wobei $\pi_H: \O_I \to \O_H$ wiederum die Projektionsabbildung. +\end{proposition} +\begin{proof} + $\nearrow$ Schilling Maß und Integral, Satz 17.4. +\end{proof} +Sind auf den Wahrscheinlichkeitsräumen $(\O_i , \F_i , \P_i), i \in I$, nun Zufallsvariablen $X_i: \O_i \to E$ gegeben, so definieren wir wie im Satz von Kolmogorov (\propref{prop:Kolmo}) +\begin{align*} + (\O, \F, \P) := (\O_I, \F_I, \P_I = \bigotimes_{i \in I}) \mit \omega = (\omega_i)_{i \in I} + \intertext{und wie im endlichen Fall} + X'_i : \O \to E \mit X'_i (\omega) = X_i (\omega_i). +\end{align*} +Da die Unabhängigkeit der Zufallsvariablen über endliche Teilfamilien definiert ist, folgt diese wie im endlichen Fall. +\subsection*{Faltungen} +Seien $X, Y$ zwei reelle und unabhängige Zufallsvariablen mit +\begin{align*} + X \sim \P_X \und Y \sim \P_Y +\end{align*} +Dann hat $(X,Y)$ die Verteilung $\P_X \otimes \P_Y$ auf $\R^2$. Andernfalls ist auch $X+Y$ eine reelle Zufallsvariable, dann +\begin{align*} + X + Y = A(X,Y) \mit A: \R^2 \to \R: (x,y) \mapsto x + y. +\end{align*} +$A$ ist stetig, also messbar. Die Verteilung von $X+Y$ ist dann $(\P_X \otimes \P_Y)\circ A^{-1}$ +\begin{definition}[Faltung] + Seien $\P_1 , \P_2$ Wahrscheinlichkeitsmaße auf $(\Rn, \borel(\Rn))$. Das durch + \begin{align*} + \P_1 \ast \P_2(F) = \iint \indi_F (x+y) \P_1 (\d x)\P_2 (\d y) + \end{align*} + definierte Wahrscheinlichkeitsmaß $\P_1 \ast \P_2 = (\P_1 \otimes \P_2) \circ A^{-1}$ auf $(\Rn, \borel(\Rn))$ heißt \begriff{Faltung} von $\P_1$ und $\P_2$. +\end{definition} \ No newline at end of file