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| -rw-r--r-- | Makefile | 19 | ||||
| -rw-r--r-- | WrStat.tex | 435 | ||||
| -rw-r--r-- | build/WrStat.pdf | bin | 0 -> 52135 bytes | |||
| -rw-r--r-- | tex/Makefile.inc | 1 | ||||
| -rw-r--r-- | tex/docstyle.sty | 102 |
5 files changed, 557 insertions, 0 deletions
diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..6346d18 --- /dev/null +++ b/Makefile @@ -0,0 +1,19 @@ +TEX := xelatex +TEXARGS := --output-directory=build --halt-on-error + +DOCNAME := WrStat +SOURCES := $(DOCNAME).tex + +include tex/Makefile.inc + +.PHONY: clean +all: build/$(DOCNAME).pdf + +clean: + @rm -rfv build + +build/$(DOCNAME).pdf : $(SOURCES) + mkdir -p build + $(TEX) $(TEXARGS) $< + $(TEX) $(TEXARGS) $< + diff --git a/WrStat.tex b/WrStat.tex new file mode 100644 index 0000000..fe4efa6 --- /dev/null +++ b/WrStat.tex @@ -0,0 +1,435 @@ +\documentclass[a4paper]{article} + +% +% Packages +% + +%% styling for this document +\usepackage{tex/docstyle} + +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{dsfont} + +\usepackage{graphicx} + +\usepackage{tabularx} +\usepackage{multirow} +\usepackage{multicol} +\usepackage{booktabs} +\usepackage{colortbl} +\usepackage{array} + +\usepackage{enumitem} +\usepackage{parskip} + +\usepackage{polyglossia} +\setmainlanguage[variant=swiss]{german} + + +% +% Metadata +% + +\title{} +\author{ + Naoki Pross%\footnote{University of Applied Sciences of Eastern Switzerland} +} +\date{} + +% +% Document +% + + +\renewcommand{\P}[1]{\mathrm{P}(#1)} +\newcommand{\given}{\,|\,} +\newcommand{\E}[1]{\mathrm{E}(#1)} +\newcommand{\Var}[1]{\mathrm{Var}(#1)} +\newcommand{\Cov}[1]{\mathrm{Cov}(#1)} +\DeclareMathOperator{\med}{\mathrm{med}} + +\begin{document} + +{\bfseries \huge \noindent + Formelblatt --- Wahrscheinlichkeit und Statistik +} + +\vspace{5mm} + +\renewcommand{\arraystretch}{2} + +\aboverulesep=0pt +\belowrulesep=0pt + +\rowcolors{3}{lightgray!20}{white} + +\noindent +\begin{tabularx}{\linewidth}{% + | >{\columncolor{black}\cellcolor{black}}p{3mm} m{10.5cm} X | + } + \hline + & \textbf{Produktregel} \newline \(k\) Positionen m\"ussen unabh\"angig von einadner markiert werden, wobai \(n_i\) verschiedene Markierungen zur Verf\"ugung stehen. + & \(\displaystyle n_1 n_2\cdots n_k = \prod_{i=1}^k n_i \) + \\ + + & \textbf{Permutation} \newline Auf wie viele Arten lassen sich \(n\) verschiedene Objekte anordnen? + & \(\displaystyle P_n = n(n-1)(n-2)\cdots1 = n! \) + \\ + + & \textbf{Kombination} \newline Auf wie vielen Arten kann man \(k\) aus \(n\) verschiedenen Objekte ausw\"ahlen? + & \(\displaystyle C_n^k = {n \choose k} = \frac{n!}{k!(n-k)!}\) + \\ + + \multirow{-7}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries Kombinatorik} + } + } + & \textbf{Variation} \newline Auf wie viele Arten kann man \(k\) mal unter \(n\) verschiedenen Objekten ausw\"ahlen? + & \(\displaystyle V_{n,k} = n^k \) + \\ + \hline +\end{tabularx} + +\vspace{3mm} + +\noindent +\begin{tabularx}{\linewidth}{% + | >{\cellcolor{black}}p{3mm} m{5cm} X | + } + + \hline + + & \textbf{Ereignis} + & + \( + \Omega = (\text{sicheres Ereignis}) \quad + \emptyset = (\text{unm\"ogliches Ereignis}) \quad + A,B \subseteq \Omega + \) + \newline + \( + A \cap B = (A \text{ und } B) \quad + A \cup B = (A \text{ oder } B) \quad + \bar{A} = \Omega \setminus A = (\text{nicht } A) + \) + \\[5pt] + + & \textbf{Wahrscheinlichkeit} + & \(\displaystyle + \mathrm{P}: \Omega \to [0;1] \quad + \P{\emptyset} = 0 \quad + \P{\Omega} = 1 \quad + \P{\bar{A}} = 1 - \P{A} + \) \newline + \(A\) und \(B\) unabh\"angig \(\iff \P{A \cap B} = \P{A}\cdot\P{B}\) + \\[5pt] + + & \textbf{Bedingte Wahrscheinlichkeit} \newline Wahrscheinlichkeit von \(A\) wenn \(B\) bereits eingetreten ist + & \(\displaystyle + \P{A \given B} = \frac{\P{A \cap B}}{\P{B}} + \stackrel{\text{\footnotesize unabh.}}{=} \P{A} \qquad + \P{\bar{A} \given B} = 1 - \P{A \given B} + \) + \\[5pt] + + & \textbf{Totale Wahrscheinlichkeit} + & \(\displaystyle + \P{A} = \sum_i \P{A \given B_i} \qquad + A \subset \bigcup_i B_i + \) + \\[8pt] + + & \textbf{Satz von Bayes} + & \( + \P{A \given B} \cdot \P{B} = \P{B \given A} \cdot \P{A} = \P{A \cap B} + \) + \\[5pt] + + %% TODO: fix this dimexpr + & \multicolumn{2}{p{\dimexpr\linewidth-12mm} | }{ + \textbf{Experimente} \newline + In einem Laplace Experiment haben alle Elementarereignisse die gleiche Wahrscheinlichkeit. In einem Bernoulli Experiment es gibt nur 2 Ereignisse \(A\) und \(\bar{A}\) mit Wahrscheinlichkeiten \(p\) und \(1-p\). + } + \\[5pt] + + & \textbf{Zufallsvariable} + & \( + X : \Omega \to U \subseteq \mathbb{R} \quad + \text{Ereignisse} \subset \Omega \,\text{ wie }\, \{ X = k \}, \{ X \leq x \}, \{ X > x \} + \) + \\[5pt] + + & \textbf{Erwartungswert} + & \(\displaystyle + \E{X} = \sum_{x \in U} x\cdot \P{X = x} \qquad + \E{X + Y} = \E{X} + \E{Y} + \) + \newline + \( + \E{\lambda X} = \lambda \E{X} \qquad + \E{XY} \stackrel{\text{\footnotesize unabh.}}{=} \E{X}\E{Y} + \) + \\[5pt] + + & \textbf{Varianz} \newline Mass f\"ur Streuung der Werte % \newline Quadratische Abweichung + & \(\Var{X} = \E{(X - \E{X})^2} = \E{X^2} - \E{X}^2\) \newline + \( + \Var{\lambda X} = \lambda^2 \Var{X} \qquad + \Var{X + Y} \stackrel{\text{\footnotesize unabh.}}{=} \Var{X} + \Var{Y} + \) + \\[5pt] + + & \textbf{Covarianz} + & \(\Cov{X,Y} = \E{XY} - \E{X}\E{Y} \stackrel{\text{\footnotesize unabh.}}{=} 0\) + \\[5pt] + + \multirow{-21}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries Wahrscheinlichkeit} + } + } + & \textbf{Satz von Tschebyscheff} + & + \\[5pt] + \hline +\end{tabularx} + +\vspace{3mm} + +\noindent +\begin{tabularx}{\linewidth}{% + | >{\cellcolor{black}}p{3mm} m{4.5cm} X | + } + \hline + & \textbf{Verteilungsfunktion} + & + ZV ist verteilt \(X \sim \mathcal{V}\) mit \(F : \mathbb{R} \to [0,1]\) monoton steigend \newline + \( + F(x) = \P{X \leq x} \qquad + F(x\to\infty) = 1\) \qquad + \(F(x\to-\infty) = 0 + \) + \\[8pt] + + & \textbf{Median} + & \(\displaystyle \med X = \inf \left\{ x : F(x) = 0.5 \right\} \) + \\[4pt] + + & \textbf{Dichtefunktion} + & \(\displaystyle + \varphi(x) = \frac{dF}{dx} \qquad + \P{a \leq X \leq b} = \int_a^b \varphi \,dx \qquad + 1 = \int_\mathbb{R} \varphi \,dx + \) + \\[8pt] + + & \textbf{Erwartungswert} + & \(\displaystyle + \E{X} = \int_\mathbb{R} x \varphi(x) \,dx \qquad + \E{X^n} = \int_\mathbb{R} x^n \varphi(x) \, dx + \) + \\[8pt] + + & \textbf{Variablentransformation} + & \(\displaystyle + Y = g(X) \qquad + \varphi_Y = \frac{\varphi_X}{g'} \circ g^{-1} + \) + \\[8pt] + + & \textbf{Standardisierung} + & \(\displaystyle + X \sim \mathcal{N}(\mu, \sigma) \qquad + Z = \frac{X - \mu}{\sigma} \sim \mathcal{N}(0,1) + \) + \\[8pt] + + \multirow{-9}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries W'keitsverteliung} + } + } + & \textbf{Rechenregeln} + & + \\[8pt] + \hline +\end{tabularx} + +\vspace{3mm} + +{ +\noindent +\renewcommand{\arraystretch}{2.1} +\begin{tabularx}{\linewidth}{% + | >{\cellcolor{black}}p{3mm} m{4.5cm} X c c c | + } + + \hline + & \bfseries Name + & \(X \sim\) + & \(\varphi(x) \text{ oder } \P{X = k}\) + & \(\E{X}\) + & \(\Var{X}\) + \\[5pt] + \hline + + & \textbf{Gleichverteilung} \newline Laplace Experimente + & \(\displaystyle \mathcal{U}(a,b)\) + & \(\displaystyle \frac{1}{b-a} \cdot \mathds{1}_{[a,b]} \) + & \(\displaystyle \frac{a + b}{2}\) + & \(\displaystyle \frac{(b - a)^2}{12} \) + \\[5pt] + + & \textbf{Exponentialverteilung} \newline Halbwertszeit \(t_\frac{1}{2} = \log(2)/a\) + & \(\displaystyle \mathcal{E}(a)\) + & \(\displaystyle ae^{-ax} \cdot \mathds{1}_{[0,\infty)} \) + & \(\displaystyle \frac{1}{a}\) + & \(\displaystyle \frac{1}{a^2}\) + \\[5pt] + + & \textbf{Normalverteilung} \newline Viele unabh. ZV + & \(\displaystyle \mathcal{N}(\mu, \sigma)\) + & \(\displaystyle \frac{1}{\sigma \sqrt{2\pi}} \cdot e^{-(x-\mu)^2 / 2\sigma^2} \) + & \(\displaystyle \mu\) + & \(\displaystyle \sigma^2 \) + \\[5pt] + + & \textbf{Potenzverteilung} \newline Pareto Verteilung + & \(\displaystyle \mathrm{Pow}(x_\textrm{m}, \alpha)\) + & + % \(\displaystyle \frac{\alpha - 1}{x_\textrm{m}} \left( + % \frac{x}{x_\textrm{m}} + % \right)^{-\alpha} \mathds{1}_{[x_\textrm{m},\infty)}\) + & \(\displaystyle x_\textrm{m} \cdot \frac{\alpha - 1}{\alpha - 2}\) + & + % \(\displaystyle \left( + % \frac{\alpha - 1}{\alpha - 3} - \left( + % \frac{\alpha - 1}{\alpha - 2} + % \right)^2 + % \right) x_\textrm{m}^2\) + \\[5pt] + + & \textbf{Chi--Quadrat V.} \newline F\"ur das \(\chi^2\) Test + & \(\displaystyle \mathcal{X}^2(k)\) + & + & + & + \\[5pt] + + \hline + + & \textbf{Geometrische V.} + & \(\displaystyle \mathcal{G}(p)\) + & \(\displaystyle p(1-p)^k \) + & \(\displaystyle \frac{1}{p}\) + & \(\displaystyle \frac{1-p}{p^2}\) + \\[5pt] + + & \textbf{Hypergeometrische V.} + & \(\displaystyle \mathcal{H}(N,R,n)\) + & \({R \choose k}{N-R \choose n-k} / {N \choose n} \) + & \(\displaystyle \frac{nR}{N}\) + & \(\displaystyle \frac{nR}{N} \left(1 - \frac{R}{N}\right) \frac{N - n}{N - 1}\) + \\[5pt] + + & \textbf{Poissonverteilung} \newline Seltener Ereignisse + & \(\displaystyle \mathcal{P}(\lambda)\) + & \(\displaystyle \frac{\lambda^k}{k!} e^{-\lambda} \) + & \(\displaystyle \lambda\) + & \(\displaystyle \lambda\) + \\[5pt] + + \multirow{-12}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries Katalog von W'keitsverteliungen} + } + } + & \textbf{Binomialverteilung} \newline Bernoulli Experimente \newline + & \(\displaystyle \mathcal{B}(n,p)\) + & \(\displaystyle {n \choose k} p^k (1 - p)^{n - k} \) + & \(\displaystyle np\) + & \(\displaystyle np(1 - p) \) + \\[5pt] + + %% TODO: fix this dimexpr and manual cell color + & + \multicolumn{5}{p{\dimexpr\linewidth-12mm} | }{ + % \cellcolor{lightgray!20} + \cellcolor{white} + F\"ur grosse \(n\) wird \(\mathcal{B}(n,p)\) + \(\displaystyle \approx \mathcal{N}\left(\mu = np, \sigma = \sqrt{np(1-p)}\right) \) + und f\"ur kleine \(p\) (selten) ist + \(\displaystyle \approx \mathcal{P}\left(\lambda = np\right) \). + } + \\[4pt] + + \hline +\end{tabularx} +} + +\vspace{3mm} + +\noindent +\setlength{\parskip}{55pt} +\begin{tabularx}{\linewidth}{% + | >{\cellcolor{black}}p{3mm} m{4.5cm} X | + } + \hline + + + & \textbf{Regression} \newline Lineares Modell + & \(\displaystyle + \text{ZV } X, Y \qquad + y \approx ax + b \qquad + a = \frac{\Cov{X,Y}}{\Var{X}} \qquad + b = \E{Y} - a\E{X} + \) + \\ + & \textbf{Regressionskoeffizient} + & \(\displaystyle + r = \frac{\Cov{X,Y}}{\sqrt{\Var{X}\Var{Y}}} \qquad + r^2 \approx 1 \implies \text{gute Approx.} + \) + \\ + + &&\\ + &&\\ + &&\\ + &&\\ + + \multirow{-5}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries Sch\"atzen} + } + } + & \textbf{} + & + \\ + \hline +\end{tabularx} + +\noindent +\begin{tabularx}{\linewidth}{% + | >{\cellcolor{black}}p{3mm} m{4.5cm} X | + } + \hline + + &&\\ + &&\\ + &&\\ + &&\\ + + \multirow{-5}{*}{\centering + \rotatebox[origin = c]{90}{ + \textcolor{white}{\bfseries Hypotesentesten} + } + } + & \textbf{} + & + \\ + \hline +\end{tabularx} + +\end{document} diff --git a/build/WrStat.pdf b/build/WrStat.pdf Binary files differnew file mode 100644 index 0000000..077f44d --- /dev/null +++ b/build/WrStat.pdf diff --git a/tex/Makefile.inc b/tex/Makefile.inc new file mode 100644 index 0000000..7492feb --- /dev/null +++ b/tex/Makefile.inc @@ -0,0 +1 @@ +SOURCES += tex/docstyle.sty diff --git a/tex/docstyle.sty b/tex/docstyle.sty new file mode 100644 index 0000000..ad4572c --- /dev/null +++ b/tex/docstyle.sty @@ -0,0 +1,102 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{tex/docstyle}[2021/02/26 v0.1 Styling for a Sans-Serif report] + +%% Margins +\RequirePackage{geometry} +\newgeometry{margin=2cm} + +%% Set math font to sans-serif +%% Note: must be set before text font +% \RequirePackage{cmbright} +% \RequirePackage{sfmath} + +%% Set font to sans-serif +\RequirePackage{fontspec} +\PassOptionsToPackage{ + usefilenames, + DefaultFeatures = { + Ligatures = Common + } +}{plex-otf} +\RequirePackage{roboto} +\renewcommand*{\familydefault}{\sfdefault} + +%% Headers and footers +\RequirePackage{fancyhdr} +\fancypagestyle{docstyle}{ + \fancyhf{} % clear currrent values + \fancyhead[L]{\itshape\leftmark} + \fancyhead[R]{} + \fancyfoot[C]{\thepage} + \renewcommand{\headrulewidth}{0pt} +} +\pagestyle{docstyle} % apply style + +%% Title style +\RequirePackage{titling} +\setlength{\droptitle}{-1cm} +\pretitle{ + \begin{flushleft} + \bfseries\huge +} +\posttitle{ + \end{flushleft} + \large +} +\preauthor{ + \begin{flushleft} + \large +} +\postauthor{ + \end{flushleft} +} +\predate{ + \begin{flushleft} +} +\postdate{ + \par\end{flushleft} +} + +%% Link colors +\PassOptionsToPackage{table}{xcolor} +\RequirePackage{xcolor} +\PassOptionsToPackage{ + plainpages=false, + pdfpagelabels, + pdfusetitle, + colorlinks = true, + linkcolor = darkgray!50!black, + urlcolor = blue!60!black, + citecolor = black, + anchorcolor = black +}{hyperref} +\RequirePackage{hyperref} + +%% Source code listings +\RequirePackage{listings} +\lstset{ + belowcaptionskip=\baselineskip, + breaklines=true, + frame=none, + inputencoding=utf8, + % margin + xleftmargin=\parindent, + % numbers + numbers=left, + numbersep=5pt, + numberstyle=\ttfamily\footnotesize\color{gray}, + % background + backgroundcolor=\color{white}, + showstringspaces=false, + % default language + language=[LaTeX]TeX, + % break long lines, and show an arrow where the line was broken + breaklines=true, + % postbreak=\mbox{\textcolor{blue!60!black}{$\hookrightarrow$}\space}, + % font + basicstyle=\ttfamily\small, + identifierstyle=\color{black}, + keywordstyle=\color{blue!60!black}, + commentstyle=\color{red!60!black}, + stringstyle=\color{orange!60!black}, +} |