Formelsammlung komplett

Wahrscheinlichkeitstheorie_und_Statistik/Formelsammlung.tex

@@ -13,7 +13,7 @@
 \usepackage[
 	pdftitle={Formelsammlung Wahrscheinlichkeitstheorie und Statistik},
 	pdfsubject={Formelsammlung "Wahrscheinlichkeitstheorie und Statistik"},
-	pdfauthor={},
+	pdfauthor={Friesen und Battermann},
 	pdfkeywords={Wahrscheinlichkeitstheorie und Statistik, Formelsammlung},
 	pdfborder={0 0 0}
 ]{hyperref}
@@ -44,7 +44,7 @@
 \fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
 \fancyhead[L]{Formelsammlung} %zentrierte Kopfzeile
 \fancyhead[C]{Wahrscheinlichkeitstheorie und Statistik} %Kopfzeile links
-\fancyhead[R]{WS 2011/2012} %Kopfzeile rechts
+\fancyhead[R]{Seite \thepage\ von \pageref{LastPage}} %Kopfzeile rechts
 \renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie
 
 \newcommand{\spa}{\hspace*{4mm}} 
@@ -56,32 +56,27 @@
 
 
 \title{Formelsammlung Wahrscheinlichkeitstheorie und Statistik}
-\author{--}
-\date{3. Semester}
 
 \begin{document}
 	\pagestyle{fancy}
 	\setcounter{page}{1}
 
-	\section*{Wahrscheinlichkeitstheorie und Statistik (Formeln)}
-	\addcontentsline{toc}{section}{Wahrscheinlichkeitstheorie und Statistik (Formeln)}
-
-	\subsection*{Eigenschaften und Rechenregeln für Wahrscheinlichkeit}
-	\addcontentsline{toc}{subsection}{Eigenschaften und Rechenregeln für Wahrscheinlichkeit}
+	\section*{Eigenschaften und Rechenregeln für Wahrscheinlichkeit}
+	\addcontentsline{toc}{section}{Eigenschaften und Rechenregeln für Wahrscheinlichkeit}
 
 	\begin{math}
-		P(\omega) = 1, P(\phi) = 0 \text{ für } A: 0 \le P(A) \le 1\\
+		P(\Omega) = 1, P(\phi) = 0 \text{ für } A: 0 \le P(A) \le 1\\
 		\overline{A} = A^C \text{ mit } P(\overline{A}) = 1-P(A) \text{ (Gegenereignis)}\\
 		P(A\cup B) = P(A) + P(B) - P(A\cap B) \text{ (Vereinigung)}\\
 		P(A\cup B) = P(A) + P(B) \text{ (Wenn A und B \underline{disjunkt})}
 	\end{math}
 
-	\subsection*{Berechnung von Wahrscheinlichkeit mit Kombinatorik}
-	\addcontentsline{toc}{subsection}{Berechnung von Wahrscheinlichkeit mit Kombinatorik}
+	\section*{Berechnung von Wahrscheinlichkeit mit Kombinatorik}
+	\addcontentsline{toc}{section}{Berechnung von Wahrscheinlichkeit mit Kombinatorik}
 
 	\begin{itemize}
 		\item[1.] Fall:\\
-			Ziehen mit Zurücklegen, mit Reihenfolge: \( n^k \quad |\omega| = n^k\)
+			Ziehen mit Zurücklegen, mit Reihenfolge: \( n^k \quad |\Omega| = n^k\)
 		\item[2.] Fall:\\
 			Ziehen ohne Zurücklegen, mit Reihenfolge: \( \frac{n!}{(n-k)!} = n^{\underline{k}} \)
 		\item[3.] Fall:\\
@@ -90,15 +85,15 @@
 			Ziehen mit Zurücklegen, ohne Reihenfolge: \( \binom{n+k-1}{k} \overset{\text{da doof}}{\Rightarrow} n^k \text{ 1. Fall}\)
 	\end{itemize}
 
-	\subsection*{Binomialkoeffizient/Multinomialkoeffizient}
-	\addcontentsline{toc}{subsection}{Binomialkoeffizient/Multinomialkoeffizient}
+	\section*{Binomialkoeffizient/Multinomialkoeffizient}
+	\addcontentsline{toc}{section}{Binomialkoeffizient/Multinomialkoeffizient}
 
 	Bi: \(\binom{n}{k} = \frac{n!}{(n-k)!*k!} \)
 
 	Multi: \( \frac{n!}{k_1! * k_2! * … * k_l!} \) mit \( k_1 + k_2 + … + k_l = n \)
 
-	\subsection*{Bedingte Wahrscheinlichkeit und Unabhängigkeit}
-	\addcontentsline{toc}{subsection}{Bedingte Wahrscheinlichkeit und Unabhängigkeit}
+	\section*{Bedingte Wahrscheinlichkeit und Unabhängigkeit}
+	\addcontentsline{toc}{section}{Bedingte Wahrscheinlichkeit und Unabhängigkeit}
 
 	\(P(B\mid A) = \frac{P(A\cap B)}{P(A)}\) (wenn \( P(A) \ne 0 \) bedingte Wahrscheinlichkeit und Bedingung)
 
@@ -106,17 +101,198 @@
 
 	\( P(A\cap B) = P(A) * P(B) \) (Wenn \(A,B\) unabhängig von einander)
 
-	\subsection*{Totale Wahrscheinlichkeit}
-	\addcontentsline{toc}{subsection}{Totale Wahrscheinlichkeit}
+	\section*{Totale Wahrscheinlichkeit}
+	\addcontentsline{toc}{section}{Totale Wahrscheinlichkeit}
 
 	Satz: \( P(A) = P(A\mid B_1) * P(B_1) + P(A\mid B_2) * P(B_2) + … + P(A\mid B_k) * P(B_k) \)
 
-	\subsection*{Zufallsvariablen}
-	\addcontentsline{toc}{subsection}{Zufallsvariablen}
+	\section*{Zufallsvariablen}
+	\addcontentsline{toc}{section}{Zufallsvariablen}
 
 	Eine Zufallsvariable \(X\) ist eine Funktion \(X: \omega \to \mathbb R\)\\
 	Die Verteilung einer Zufallsvariable gibt an, mit welchen Wahrscheinlichkeiten die Zufallsvariable ihre möglichen Werte annimmt.
 
+	\subsection*{Diskrete Zufallsvariable}
+	\addcontentsline{toc}{subsection}{Diskrete Zufallsvariable}
+
+	Diskret: \(X\) nimmt nur einzelne Werte mit bestimmten Wahrscheinlichkeiten an
+
+	\subsubsection*{Erwartungswert (Mittelwert)}
+	\addcontentsline{toc}{subsubsection}{Erwartungswert (Mittelwert)}
+
+	\( M = E(X) = \sum\limits_i x_i * p_i \) mit \( (i=1,2,3,…) \)
+
+	\subsubsection*{Streuung (Varianz)}
+	\addcontentsline{toc}{subsubsection}{Streuung (Varianz)}
+
+	\( \sigma^2 = V(X) = Var(X) = \sum\limits_i (x_i-\mu)^2 * p_i \to\) mittlerer Quadratischer Abstand
+
+	Standardabweichung \(\sigma\) ist \( \sigma = \sqrt{\sigma^2} \)
+
+	Wenn \( \sigma = 0 \Rightarrow x = \mu \) mit Wahrscheinlichkeit 1
+
+	\( \sigma^2 = \sum\limits_i x_i * p_i - \mu^2 \) (einfachere Formel)
+
+	Variationskoeffizient \(\frac{\sigma}{\mu}\) in (\%)
+
+	\section*{Verteilung und Verteilungsfunktion von Zufallsvariablen}
+	\addcontentsline{toc}{section}{Verteilung und Verteilungsfunktion von Zufallsvariablen}
+
+	Funktion \(F: \mathbb R \to [0,1] \) mit \( F(x) = P(X \le x) \)
+
+	\subsection*{Eigenschaften von \(F(x)\)}
+	\addcontentsline{toc}{subsection}{Eigenschaften von \(F(x)\)}
+
+	\begin{itemize}
+		\item \( \lim\limits_{x\to-\infty} F(x) = 0 \) und \( \lim\limits_{x\to\infty} F(x) = 1 \)
+		\item \(F(x)\) ist monoton Wachsend (nicht unbedingt streng)
+		\item \(F(x)\) ist rechtsseitig stetig \( (F(x) ? P(X\le c) )\)
+	\end{itemize}
+
+	\subsection*{Verteilungsfunktion}
+	\addcontentsline{toc}{subsection}{Verteilungsfunktion}
+
+	\begin{itemize}
+		\item eine Treppenfunktion \(\to\) bei diskreten Verteilungen
+		\item eine durchgehend stetige Funktion \(\to\) Keine Sprünge
+	\end{itemize}
+
+	\section*{Stetige Verteilung}
+	\addcontentsline{toc}{section}{Stetige Verteilung}
+
+	\begin{itemize}
+		\item stetige Funktion \(F(x)\)
+		\item \(P(X=x) = 0 \forall x \in\mathbb R \) (\(\Rightarrow\) Keine Sprünge)
+	\end{itemize}
+	Stetige Stetige Verteilungsfunktionen besitzen eine Dichtefunktion \(f(x)\) mit
+	\begin{itemize}
+		\item \(f(x) = F'(x)\)
+	\end{itemize}
+
+	\subsection*{Eigenschaften Dichtefunktion}
+	\addcontentsline{toc}{subsection}{Eigenschaften Dichtefunktion}
+
+	\begin{itemize}
+		\item \( f: \mathbb R \to \left[0,\infty\right)\), also \(f(x) \ge 0 \forall x \in \mathbb R \)
+		\item \( \int\limits_{-\infty}^\infty f(x) \mathrm dx = 1\)
+	\end{itemize}
+
+	\section*{Diskrete Verteilung}
+	\addcontentsline{toc}{section}{Diskrete Verteilung}
+
+	Gleichverteilung: \( P(X=x_1) = P(X=x_2) = … = P(X=x_n) = \frac1n \)
+
+	\section*{Geometrische Verteilung}
+	\addcontentsline{toc}{section}{Geometrische Verteilung}
+
+	\(X\) ist geometrisch verteilt mit Parameter \(p\): \(P(X=k) = (1-p)^{k-1}*p \)
+
+	\section*{Binomialverteilung}
+	\addcontentsline{toc}{section}{Binomialverteilung}
+
+	Ein Zufallsexperiment wird \(n\)-mal unabhängig voneinander wiederholt\\
+	\( P(X=k) = \binom{n}{k} * p^k *(1-p)^{n-k} \)
+	\begin{itemize}
+		\item \(\mu = \sum\limits_{k=0}^n k * \binom{n}{k} *p^k * (1-p)^{n-k} = n*p \)
+		\item \(\sigma^2 = n*p*(1-p) \)
+	\end{itemize}
+	für \( p = 0,5 \) ist die Verteilung Symmetrisch
+
+	\section*{Gaußsche Normalverteilung}
+	\addcontentsline{toc}{section}{Gaußsche Normalverteilung}
+
+	\(f(x) = \frac{1}{\sqrt{2\pi} * \sigma} *e^{-\frac12*(\frac{\pi-\mu}{\sigma})^2} \)
+
+	Veränderung von \(\mu\to\) Verschiebung ind \(x\)-Richtung
+
+	Veränderung von \(\sigma\to\) Stauchung/Dehnung in der Breite/Höhe
+
+	Standardnormalverteilung: \( N(0,1): F(x) = \frac{1}{\sqrt{2\pi}} * e^{-\frac12x^2} \)
+
+	\section*{Erwartungswert und Varianz stetiger Verteilung}
+	\addcontentsline{toc}{section}{Erwartungswert und Varianz stetiger Verteilung}
+
+	\underline{diskret:} \( \mu = \sum\limits_i x_i * p_i \quad \sigma^2 = \sum\limits_i x_i^2 * p_i -\mu^2 \)
+
+	\underline{stetig:} \( \mu = \int\limits_{-\infty}^\infty x * f(x) \mathrm dx \quad \sigma^2 = \int\limits_{-\infty}^\infty x^2 * f(x) \mathrm dx - \mu^2 \)
+
+	\section*{Normalverteilung}
+	\addcontentsline{toc}{section}{Normalverteilung}
+
+	Wenn \(f(x) \in X\), dann \(X\) ist \(N(\mu,\sigma)\) verteilt.
+
+	Es gilt: \( E(X) = \mu, Var(X) = \sigma^2 \)
+
+	\(\Rightarrow X\) heißt standardnormalverteilt, wenn \(\mu = 0, \sigma = 1 \Rightarrow f(x) = \frac{1}{\sqrt{2\pi}}*e^{-\frac12x^2} \)
+
+	Verteilungsfunktion \(F(x) = \phi(x) = \phi_{0,1}(x) \to \) hierfür: Tabellenwerte
+
+	\includegraphics{bilder/fosa_dichtefunktion.eps}
+
+	\section*{Schätzungen und Tests}
+	\addcontentsline{toc}{section}{Schätzungen und Tests}
+
+	\subsection*{Stichproben}
+	\addcontentsline{toc}{subsection}{Stichproben}
+
+	\begin{itemize}
+		\item arithmetisches Mittel (Durchschnitt)\\
+			\( \overline{x} = \frac1n * (x_1+x_2+…+x_n) = \frac1n \sum\limits_{i=1}^n x_i \)
+		\item Schätzung für \(\mu\)\\
+			\( \overline{x} = \frac1n * (x_1+x_2+…+x_n) \)
+		\item Schätzwert für \(\sigma\)\\
+			\( s^2 = \frac{1}{n-1} * (x_1^2 + x_2^2 + … + x_n^2 ) - \frac{n}{n-1} * \overline{x}^2 = \sqrt{s_{n-1}^2} \)
+	\end{itemize}
+
+	\subsection*{Zentraler Grenzwertsatz}
+	\addcontentsline{toc}{subsection}{Zentraler Grenzwertsatz}
+
+	\( \overline{x} = \frac1n * (x_1+x_2+…+x_n) \)
+
+	\( \frac{\overline{x}-\mu}{\sigma} * \sqrt{n} \) ist für große \(n\) näherungsweise \(N(0,1)\) verteilt.
+
+	\subsection*{Tests}
+	\addcontentsline{toc}{subsection}{Tests}
+
+	\( H_0: \mu = \mu_0 \Rightarrow \) zweiseitiger Test
+
+	\( H_0: \mu \le \mu_0 \) oder \( H_0: \mu \ge \mu_0 \Rightarrow \) einseitige Tests
+
+	\( T= \frac{\overline{x}-\mu_0}{\sigma} * \sqrt{n} \)
+
+	\includegraphics{bilder/fosa_dichtefunktion_zwei.eps}\\
+	zweiseitiger Test
+
+	Testentscheidung: \(H_0\) wird \(\begin{cases} \text{angenommen} &, -c \le T \le c \\ \text{abgelehnt} &, T>c,T<-c \end{cases} \)
+
+	\underline{Bestimmung von \(c\)}
+
+	\( \phi(c) = 1-\frac\alpha2,\quad c=\phi^{-1}(1-\frac\alpha2) \) (rückwärts)
+
+	\underline{Einseitige Tests}
+
+	\(H_0: \mu \le \mu_0 \) (vorgegebener Maximalwert)
+
+	\includegraphics{bilder/fosa_dichtefunktion_max.eps}
+
+	\(H_0\) wird \(\begin{cases} \text{angenommen} &, T \le c \\ \text{abgelehnt} &, T>c \end{cases} \)
+
+	\(H_0: \mu \ge \mu_0 \) (vorgegebener Minimalwert)
+
+	\includegraphics{bilder/fosa_dichtefunktion_min.eps}
+
+	\(H_0\) wird \(\begin{cases} \text{angenommen} &, T \ge c \\ \text{abgelehnt} &, T<c \end{cases} \)
+
+	\subsection*{Konfidenzintervall}
+	\addcontentsline{toc}{subsection}{Konfidenzintervall}
+
+	Konfidenzintervall (Vertrauensintervall) für den Erwartungswert $\mu$ $\to$ Intervall, das um den Stichproben-Schätzwert $\overline{x}$ herumliegt, in dem $\mu$ mit Wahrscheinlichkeit \cunder{Yellow}{$1-\alpha$} liegt.
 
+	Analog zu den Tests (zweiseitig): $P(-c \le T \le c) = 1-\alpha$\\
+	$\to \underbrace{P\Big(-\phi^{-1}(1-\frac\alpha2) \le \underbrace{T}_{ \frac{\overline{x}-\mu}{\sigma} * \sqrt{n} } \le \phi^{-1}(1-\frac\alpha2) \Big) }_{\color{Orange}\text{Umformen, so dass } \mu \text{ in der Mitte steht}}$\\
+	$\to P\Big( \overline{x}-\phi^{-1}(1-\frac{\alpha}{2}) * \frac{\sigma}{\sqrt{n}}\ \le\ \mu\ \le\ \overline{x}+\phi^{-1}(1-\frac{\alpha}{2}) * \frac{\sigma}{\sqrt{n}}\Big)$\\
+	$\to$ Formel für das $(1-\alpha)$ Konfidenzintervall:\\
+	$ \Big[ \overline{x}-\phi^{-1}(1-\frac{\alpha}{2}) * \frac{\sigma}{\sqrt{n}}\ ,\ \overline{x}+\phi^{-1}(1-\frac{\alpha}{2}) * \frac{\sigma}{\sqrt{n}} \Big] $ \textcolor{Orange}{$\leftarrow$ da drin liegt $\mu$ mit Wahrscheinlichkeit $1-\alpha$}\\
+	\includegraphics{bilder/1-6_konfidenz.eps}
 
 \end{document}

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