From 8484b46ad7dc4893ee27748d25cb6b22e9b979de Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=ABl=20Leirer?= Date: Thu, 17 Nov 2022 13:05:07 +0100 Subject: [PATCH] added Standardnormalverteilungstabelle --- WrStatHS22.tex | 134 +++++++++++++++++++-------- WrStatHS22.xdv | Bin 151388 -> 235300 bytes include/NormalverteilungsTabelle.tex | 75 +++++++++++++++ 3 files changed, 168 insertions(+), 41 deletions(-) create mode 100644 include/NormalverteilungsTabelle.tex diff --git a/WrStatHS22.tex b/WrStatHS22.tex index 81854b3..623d915 100644 --- a/WrStatHS22.tex +++ b/WrStatHS22.tex @@ -19,6 +19,8 @@ \usepackage{pgfplotstable} \pgfplotsset{compat = newest} \setdefaultlanguage[variant=swiss]{german} +\usepackage{color, colortbl} +\usepackage{graphicx} %% License configuration \usepackage[ @@ -255,10 +257,12 @@ \section{Statistik} \subsubsection*{Quantil} Eine Unterteilung der Beobachtungswerte bzw. der Fläche der Dichtekurve. $p$-Quantil $p \in \mathbb{R}$ links davon ist der Anteil $p$ in \% aller Zufallswerte bzw. Der Dichtekurve. +\end{multicols} - \subsection{diskrete Zufallsvariable} +\subsection{diskrete Zufallsvariable} +\begin{multicols}{2} Eine Zufallsvariable X ist eine Funktion die einem Versuchsausgang $\omega$ einen Wert $X(\omega)$ zuordnet. @@ -271,18 +275,22 @@ \section{Statistik} $$ =\sum _{\omega \in \Omega} X(\omega) \cdot P({\omega}) = \sum _{i=1} ^{k} X(A_i) \cdot P(A_i)$$ \subsubsection*{Empirischer Erwartungswert} - $$ E(X) = \sum _i X(i) \cdot P(X=x_i) \approx \sum _i x_i \frac{n_i}{n} = \frac{x_1n_1 + x_2n_2 + \dots + x_kn_k}{n}$$ + $$ E(X) = \sum _i X(i) \cdot P(X=x_i)$$ + $$\approx \sum _i x_i \frac{n_i}{n} = \frac{x_1n_1 + x_2n_2 + \dots + x_kn_k}{n}$$ \subsubsection*{Rechenregeln} $$E(X+Y) = E(X) + E(Y)$$ $$E(\lambda X) = \lambda E(X)$$ - Vorzeichen von XY unkorreliert $\Rightarrow E(XY) = E(X)E(Y)$ - Unabhängig: $XY \geq 0 \Rightarrow E(XY) \geq 0$ oder $XY\leq 0 \Rightarrow E(XY) \leq 0$ - - X und Y sind unabhängig, wenn die Ereignisse x und y unabhängig sind. + \begin{itemize} + \item Vorzeichen von XY unkorreliert \newline $\Rightarrow E(XY) = E(X)E(Y)$ + \item Unabhängig: $XY \geq 0 \Rightarrow E(XY) \geq 0$ \newline oder $XY\leq 0 \Rightarrow E(XY) \leq 0$ + \item X und Y sind unabhängig, wenn die Ereignisse x und y unabhängig sind. + \end{itemize} +\end{multicols} - \subsection{Varianz - Streumass} +\subsection{Varianz - Streumass} +\begin{multicols}{2} Die Varianz ist die Erwartete Abweichung vom Erwartungswert. Sie misst die Streuung einer Zufallsvariable. @@ -304,7 +312,7 @@ \section{Statistik} für $|M_n -\mu| > \epsilon$ \subsubsection*{Gesetz grosser Zahlen} Angenommen die Wahrscheinlichkeit von Ereigniss A ist gleich dem Erwartungswert ($P(A) = E(X)$), dann gilt: - \\ \textbf{relative Häufigkeit:} $h_n = \frac{X_1+ \dots + X_n}{n}$, $E(h_n) = P(A)$ , $ var(h_n) = \frac{1}{n} var(X) = \frac{1}{n}P(A)(1-P(A))$ + \\ \textbf{Relative Häufigkeit:} $h_n = \frac{X_1+ \dots + X_n}{n}$, $E(h_n) = P(A)$ , $ var(h_n) = \frac{1}{n} var(X) = \frac{1}{n}P(A)(1-P(A))$ \\ \textbf{Satz von Bernoulli:} $$P(|h_n - P(A)|) \leq \frac{P(A)(1-P(A))}{n\epsilon^2} \leq \frac{1}{4n\epsilon^2} $$ für $|h_n - P(A)| > \epsilon$ @@ -313,7 +321,7 @@ \section{Statistik} \end{multicols} -\subsection*{Lineare Regression} +\subsection{Lineare Regression} \begin{multicols}{2} \begin{tikzpicture} @@ -375,56 +383,100 @@ \subsection*{Lineare Regression} \subsection{Stetige Zufallsvariable} \begin{multicols}{2} - + \subsubsection*{Verteilungsfunktion} Die Verteilungsfunktion beschreib die Wahrscheinlichkeiten der Werte einer Zufallsvariable. -$$F(x)= F_X(x) = P(X \leq x)$$ -Eigenschaften: -\begin{itemize} - \item monoton wachsend - \item $0 \leq F_X(x) \leq 1$ - \item $\lim_{x \to -\infty} F(x) = 0$ - \item $lim_{x \to \infty} F(x) = 1$ - \item $P(a < X \leq b) = F_x(b) - F_X(a)$ -\end{itemize} + $$F(x)= F_X(x) = P(X \leq x)$$ + Eigenschaften: + \begin{itemize} + \item monoton wachsend + \item $0 \leq F_X(x) \leq 1$ + \item $\lim_{x \to -\infty} F(x) = 0$ + \item $lim_{x \to \infty} F(x) = 1$ + \item $P(a < X \leq b) = F_x(b) - F_X(a)$ + \end{itemize} -\subsubsection*{Wahrscheinlichkeitsdichte:} -Die Wahrscheinlichkeitsdichte ist die Ableitung der Verteilungsfunktion. -Ihre Fläche \textbf{muss} 1 sein. + \subsubsection*{Wahrscheinlichkeitsdichte:} + Die Wahrscheinlichkeitsdichte ist die Ableitung der Verteilungsfunktion. + Ihre Fläche \textbf{muss} 1 sein. -$$\varphi(x) = \varphi_X(x) = F'_X(x)$$ -$$F_X(x) = \int \limits _{-\infty} ^{x} \varphi(\xi)d\xi$$ + $$\varphi(x) = \varphi_X(x) = F'_X(x)$$ + $$F_X(x) = \int \limits _{-\infty} ^{x} \varphi(\xi)d\xi$$ -Eigenschaften: -\begin{itemize} - \item $\varphi(x) \geq 0$ - \item $\int \limits _{-\infty} ^{\infty} \varphi(x)dx = 1$ - \item $F_X(x)$ ist die Stammfunktion. - \item $P(a < X \leq b) =\int \limits _a ^b \varphi_X(x)dx$ -\end{itemize} + Eigenschaften: + \begin{itemize} + \item $\varphi(x) \geq 0$ + \item $\int \limits _{-\infty} ^{\infty} \varphi(x)dx = 1$ + \item $F_X(x)$ ist die Stammfunktion. + \item $P(a < X \leq b) =\int \limits _a ^b \varphi_X(x)dx$ + \end{itemize} -\subsubsection{Erwartungswert:} -$$E(X) = \sum Wert \cdot Wahrscheinlichkeit = \int \limits _{-\infty} ^{\infty} x \cdot \varphi(x)dx $$ -$$E(X^2) = \int \limits _{-\infty} ^{\infty} x^2 \cdot \varphi(x) dx$$ -$$E(f(X)) = \int \limits _{-\infty} ^{\infty} f(x) \cdot \varphi(x)dx$$ + \subsubsection{Erwartungswert:} + $$E(X) = \sum Wert \cdot Wahrscheinlichkeit = \int \limits _{-\infty} ^{\infty} x \cdot \varphi(x)dx $$ + $$E(X^2) = \int \limits _{-\infty} ^{\infty} x^2 \cdot \varphi(x) dx$$ + $$E(f(X)) = \int \limits _{-\infty} ^{\infty} f(x) \cdot \varphi(x)dx$$ -\subsubsection*{Summe Stetiger Zufallsvariablen} -wemm $X$ und $Y$ unabhängig sind, dann hat die Zufallsvariable $Z = X + Y$ die Wahrscheinlichkeitsdichte: -$$\varphi_Z(z) = \varphi_X * \varphi_Y(z) = \int \limits _{-\infty} ^{\infty} \varphi_X(x) \varphi_Y(z-x)dx$$ + \subsubsection*{Summe Stetiger Zufallsvariablen} + wemm $X$ und $Y$ unabhängig sind, dann hat die Zufallsvariable $Z = X + Y$ die Wahrscheinlichkeitsdichte: + $$\varphi_Z(z) = \varphi_X * \varphi_Y(z) = \int \limits _{-\infty} ^{\infty} \varphi_X(x) \varphi_Y(z-x)dx$$ -\subsubsection*{Summe von Gleichverteilungen} -$$\varphi_X(x) = \varphi_Y(x)$$ -$$\varphi_{X+Y}(x) = \varphi_X * \varphi_Y(x) = \int \limits _{-\infty} ^{\infty} \varphi(y)\cdot \varphi(x-y)dy $$ + \subsubsection*{Summe von Gleichverteilungen} + $$\varphi_X(x) = \varphi_Y(x)$$ + $$\varphi_{X+Y}(x) = \varphi_X * \varphi_Y(x) = \int \limits _{-\infty} ^{\infty} \varphi(y)\cdot \varphi(x-y)dy $$ \end{multicols} +\subsection{Verteiungen} +\subsubsection{Normalverteilung} +X heisst Normalverteilt wenn gilt: +$$\varphi(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ + +\subsubsection{Poisson-Verteilung} +{\huge TODO} + +\subsubsection{Exponentialverteilung} +{\huge TODO} +\subsection{Binomialverteilung} +{\huge TODO} + +\subsection{Hypergeometrische Verteilung} +{\huge TODO} + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Anhang %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Anhang} +\subsection*{Formelzeichen} +\begin{itemize} + \item + \item $\bar{x} = M_n$ Mittelwert (i.d.R. Arithmetisches Mittel) + \item $G =$ Geometrisches Mittel + \item $var(\dots)$ Varianz von \dots + \item $P(\dots) =$ Wahrscheinlichkeit für Eintreffen von Ereigniss \dots + \item $E(\dots) =$ Erwartungswert von Ereignis \dots + \item + \item $\varphi(\dots =)$ Wahrscheinlichkeitsverteilung von Ereigniss \dots + \item +\end{itemize} \subsection{Taschenrechner} +\newpage +\section{Tabellen} +\subsection{Quantile der Standardnormalverteilung} + +\begin{tabular}{|c|c|c|c|c|c|c|c|} + \hline + $q$ & 0,750 & 0,800 & 0,900 & 0,950 & 0,975 & 0,990 & 0,995 \\ + \hline + $z_q$ & 0,674490 & 0,841621 & 1,281550 & 1,644850 & 1,959960 & 2,326350 & 2,575830 \\ + \hline +\end{tabular} + +\input{include/NormalverteilungsTabelle.tex} + \newpage \subsection{Berechnungstabelle Lineare Regression} diff --git a/WrStatHS22.xdv b/WrStatHS22.xdv index f47def955b38196d5ad21cd34379f8b313d52e3d..14de9fc9ae120baed9a06186026b301abf363d05 100644 GIT binary patch literal 235300 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0,51595 & 0,51994 & 0,52392 & 0,52790 & 0,53188 & 0,53586 \\ + \rowcolor{LightGray} + 0,1 & 0,53983 & 0,54380 & 0,54776 & 0,55172 & 0,55567 & 0,55962 & 0,56356 & 0,56749 & 0,57142 & 0,57535 \\ + 0,2 & 0,57926 & 0,58317 & 0,58706 & 0,59095 & 0,59483 & 0,59871 & 0,60257 & 0,60642 & 0,61026 & 0,61409 \\ + \rowcolor{LightGray} + 0,3 & 0,61791 & 0,62172 & 0,62552 & 0,62930 & 0,63307 & 0,63683 & 0,64058 & 0,64431 & 0,64803 & 0,65173 \\ + 0,4 & 0,65542 & 0,65910 & 0,66276 & 0,66640 & 0,67003 & 0,67364 & 0,67724 & 0,68082 & 0,68439 & 0,68793 \\ + \rowcolor{LightGray} + 0,5 & 0,69146 & 0,69497 & 0,69847 & 0,70194 & 0,70540 & 0,70884 & 0,71226 & 0,71566 & 0,71904 & 0,72240 \\ + 0,6 & 0,72575 & 0,72907 & 0,73237 & 0,73565 & 0,73891 & 0,74215 & 0,74537 & 0,74857 & 0,75175 & 0,75490 \\ + \rowcolor{LightGray} + 0,7 & 0,75804 & 0,76115 & 0,76424 & 0,76730 & 0,77035 & 0,77337 & 0,77637 & 0,77935 & 0,78230 & 0,78524 \\ + 0,8 & 0,78814 & 0,79103 & 0,79389 & 0,79673 & 0,79955 & 0,80234 & 0,80511 & 0,80785 & 0,81057 & 0,81327 \\ + \rowcolor{LightGray} + 0,9 & 0,81594 & 0,81859 & 0,82121 & 0,82381 & 0,82639 & 0,82894 & 0,83147 & 0,83398 & 0,83646 & 0,83891 \\ + 1,0 & 0,84134 & 0,84375 & 0,84614 & 0,84849 & 0,85083 & 0,85314 & 0,85543 & 0,85769 & 0,85993 & 0,86214 \\ + \rowcolor{LightGray} + 1,1 & 0,86433 & 0,86650 & 0,86864 & 0,87076 & 0,87286 & 0,87493 & 0,87698 & 0,87900 & 0,88100 & 0,88298 \\ + 1,2 & 0,88493 & 0,88686 & 0,88877 & 0,89065 & 0,89251 & 0,89435 & 0,89617 & 0,89796 & 0,89973 & 0,90147 \\ + \rowcolor{LightGray} + 1,3 & 0,90320 & 0,90490 & 0,90658 & 0,90824 & 0,90988 & 0,91149 & 0,91309 & 0,91466 & 0,91621 & 0,91774 \\ + 1,4 & 0,91924 & 0,92073 & 0,92220 & 0,92364 & 0,92507 & 0,92647 & 0,92785 & 0,92922 & 0,93056 & 0,93189 \\ + \rowcolor{LightGray} + 1,5 & 0,93319 & 0,93448 & 0,93574 & 0,93699 & 0,93822 & 0,93943 & 0,94062 & 0,94179 & 0,94295 & 0,94408 \\ + 1,6 & 0,94520 & 0,94630 & 0,94738 & 0,94845 & 0,94950 & 0,95053 & 0,95154 & 0,95254 & 0,95352 & 0,95449 \\ + \rowcolor{LightGray} + 1,7 & 0,95543 & 0,95637 & 0,95728 & 0,95818 & 0,95907 & 0,95994 & 0,96080 & 0,96164 & 0,96246 & 0,96327 \\ + 1,8 & 0,96407 & 0,96485 & 0,96562 & 0,96638 & 0,96712 & 0,96784 & 0,96856 & 0,96926 & 0,96995 & 0,97062 \\ + \rowcolor{LightGray} + 1,9 & 0,97128 & 0,97193 & 0,97257 & 0,97320 & 0,97381 & 0,97441 & 0,97500 & 0,97558 & 0,97615 & 0,97670 \\ + 2,0 & 0,97725 & 0,97778 & 0,97831 & 0,97882 & 0,97932 & 0,97982 & 0,98030 & 0,98077 & 0,98124 & 0,98169 \\ + \rowcolor{LightGray} + 2,1 & 0,98214 & 0,98257 & 0,98300 & 0,98341 & 0,98382 & 0,98422 & 0,98461 & 0,98500 & 0,98537 & 0,98574 \\ + 2,2 & 0,98610 & 0,98645 & 0,98679 & 0,98713 & 0,98745 & 0,98778 & 0,98809 & 0,98840 & 0,98870 & 0,98899 \\ + \rowcolor{LightGray} + 2,3 & 0,98928 & 0,98956 & 0,98983 & 0,99010 & 0,99036 & 0,99061 & 0,99086 & 0,99111 & 0,99134 & 0,99158 \\ + 2,4 & 0,99180 & 0,99202 & 0,99224 & 0,99245 & 0,99266 & 0,99286 & 0,99305 & 0,99324 & 0,99343 & 0,99361 \\ + \rowcolor{LightGray} + 2,5 & 0,99379 & 0,99396 & 0,99413 & 0,99430 & 0,99446 & 0,99461 & 0,99477 & 0,99492 & 0,99506 & 0,99520 \\ + 2,6 & 0,99534 & 0,99547 & 0,99560 & 0,99573 & 0,99585 & 0,99598 & 0,99609 & 0,99621 & 0,99632 & 0,99643 \\ + \rowcolor{LightGray} + 2,7 & 0,99653 & 0,99664 & 0,99674 & 0,99683 & 0,99693 & 0,99702 & 0,99711 & 0,99720 & 0,99728 & 0,99736 \\ + 2,8 & 0,99744 & 0,99752 & 0,99760 & 0,99767 & 0,99774 & 0,99781 & 0,99788 & 0,99795 & 0,99801 & 0,99807 \\ + \rowcolor{LightGray} + 2,9 & 0,99813 & 0,99819 & 0,99825 & 0,99831 & 0,99836 & 0,99841 & 0,99846 & 0,99851 & 0,99856 & 0,99861 \\ + 3,0 & 0,99865 & 0,99869 & 0,99874 & 0,99878 & 0,99882 & 0,99886 & 0,99889 & 0,99893 & 0,99896 & 0,99900 \\ + \rowcolor{LightGray} + 3,1 & 0,99903 & 0,99906 & 0,99910 & 0,99913 & 0,99916 & 0,99918 & 0,99921 & 0,99924 & 0,99926 & 0,99929 \\ + 3,2 & 0,99931 & 0,99934 & 0,99936 & 0,99938 & 0,99940 & 0,99942 & 0,99944 & 0,99946 & 0,99948 & 0,99950 \\ + \rowcolor{LightGray} + 3,3 & 0,99952 & 0,99953 & 0,99955 & 0,99957 & 0,99958 & 0,99960 & 0,99961 & 0,99962 & 0,99964 & 0,99965 \\ + 3,4 & 0,99966 & 0,99968 & 0,99969 & 0,99970 & 0,99971 & 0,99972 & 0,99973 & 0,99974 & 0,99975 & 0,99976 \\ + \rowcolor{LightGray} + 3,5 & 0,99977 & 0,99978 & 0,99978 & 0,99979 & 0,99980 & 0,99981 & 0,99981 & 0,99982 & 0,99983 & 0,99983 \\ + 3,6 & 0,99984 & 0,99985 & 0,99985 & 0,99986 & 0,99986 & 0,99987 & 0,99987 & 0,99988 & 0,99988 & 0,99989 \\ + \rowcolor{LightGray} + 3,7 & 0,99989 & 0,99990 & 0,99990 & 0,99990 & 0,99991 & 0,99991 & 0,99992 & 0,99992 & 0,99992 & 0,99992 \\ + 3,8 & 0,99993 & 0,99993 & 0,99993 & 0,99994 & 0,99994 & 0,99994 & 0,99994 & 0,99995 & 0,99995 & 0,99995 \\ + \rowcolor{LightGray} + 3,9 & 0,99995 & 0,99995 & 0,99996 & 0,99996 & 0,99996 & 0,99996 & 0,99996 & 0,99996 & 0,99997 & 0,99997 \\ + 4,0 & 0,99997 & 0,99997 & 0,99997 & 0,99997 & 0,99997 & 0,99997 & 0,99998 & 0,99998 & 0,99998 & 0,99998 \\ + \hline +\end{tabular} \ No newline at end of file