From b5084b78f44632f51eacd0546f43da969918e62f Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Mon, 19 Oct 2015 17:35:27 +0200
Subject: [PATCH 1/4] Added exercises

---
 statistics-fabian/assignments/day1_001.pdf    | Bin 53507 -> 0 bytes
 statistics-fabian/assignments/day1_002.pdf    | Bin 52598 -> 0 bytes
 statistics-fabian/assignments/example001.csv  |  43 ----
 statistics-fabian/assignments/example002.csv  | 186 ------------------
 statistics/exercises/UT_WBMW_Black_RGB.pdf    | Bin 0 -> 31288 bytes
 .../exercises/descriptivestatistics-01.tex    | 162 +++++++++++++++
 .../exercises/descriptivestatistics-02.tex    | 164 +++++++++++++++
 statistics/lecture/descriptivestatistics.tex  |   4 +-
 8 files changed, 328 insertions(+), 231 deletions(-)
 delete mode 100644 statistics-fabian/assignments/day1_001.pdf
 delete mode 100644 statistics-fabian/assignments/day1_002.pdf
 delete mode 100755 statistics-fabian/assignments/example001.csv
 delete mode 100755 statistics-fabian/assignments/example002.csv
 create mode 100644 statistics/exercises/UT_WBMW_Black_RGB.pdf
 create mode 100644 statistics/exercises/descriptivestatistics-01.tex
 create mode 100644 statistics/exercises/descriptivestatistics-02.tex

diff --git a/statistics-fabian/assignments/day1_001.pdf b/statistics-fabian/assignments/day1_001.pdf
deleted file mode 100644
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diff --git a/statistics-fabian/assignments/example002.csv b/statistics-fabian/assignments/example002.csv
deleted file mode 100755
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diff --git a/statistics/exercises/descriptivestatistics-01.tex b/statistics/exercises/descriptivestatistics-01.tex
new file mode 100644
index 0000000..9800e54
--- /dev/null
+++ b/statistics/exercises/descriptivestatistics-01.tex
@@ -0,0 +1,162 @@
+\documentclass[12pt,a4paper,pdftex]{exam}
+
+\usepackage[german]{babel}
+\usepackage{natbib}
+\usepackage{graphicx}
+\usepackage[small]{caption}
+\usepackage{sidecap}
+\usepackage{pslatex}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\setlength{\marginparwidth}{2cm}
+\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
+
+%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
+\pagestyle{headandfoot}
+\header{{\bfseries\large \"Ubung 1}}{{\bfseries\large Deskriptive Statistik}}{{\bfseries\large 19. Oktober, 2015}}
+\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
+jan.grewe@uni-tuebingen.de}
+\runningfooter{}{\thepage}{}
+
+\setlength{\baselineskip}{15pt}
+\setlength{\parindent}{0.0cm}
+\setlength{\parskip}{0.3cm}
+\renewcommand{\baselinestretch}{1.15}
+
+\newcommand{\qt}[1]{\textbf{#1}\\}
+\newcommand{\pref}[1]{(\ref{#1})}
+\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
+\newcommand{\code}[1]{\texttt{#1}}
+
+\newcommand{\continue}{\ifprintanswers%
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\continuepage}{\ifprintanswers%
+\newpage
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\newsolutionpage}{\ifprintanswers%
+\newpage%
+\else
+\fi}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+
+\vspace*{-6.5ex}
+\begin{center}
+\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex]
+{\large Jan Grewe, Jan Benda}\\[-3ex]
+Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
+\end{center}
+
+% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
+% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
+% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
+% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
+% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
+% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
+% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
+% (z.B. variablen\_datentypen\_mueller.m).
+
+\begin{itemize}
+\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie
+auch wirklich das macht, was sie machen soll! Teste dies mit kleinen
+Beispielen direkt in der Kommandozeile.
+\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in
+sinnvolle kleine Funktionen herunterzubrechen.
+\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
+lohnt es sich eine Funktion daraus zu schreiben!
+\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen
+Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles
+stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute
+Statistik zu bekommen.
+\item Benutze die Hilfsfunktion von matlab und das Internet, um
+herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden
+sind und was f\"ur M\"oglichkeiten sie bieten.
+\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten!
+\end{itemize}
+
+
+\begin{questions}
+
+\question \qt{Wahrscheinlichkeiten eines W\"urfels I}
+Der Computer kann auch als W\"urfel verwendet werden!
+\begin{parts}
+  \part Simuliere 10000 W\"urfe mit dem W\"urfel durch Erzeugung von
+  ganzzahligen Zufallszahlen mit den Augenzahlen $x_i = 1, 2, \ldots 6$ .
+  \part Berechne die Wahrscheinlichkeit $P(3)$
+  des Auftretens der Augenzahl drei durch Bestimmung der Anzahl der Dreien im Datensatz.\\
+  Entspricht das Ergebnis deiner Erwartung?\\
+  \"Uberpr\"ufe auch die Wahrscheinlichkeit $P(x_i)$ der anderen Zahlen.\\
+  Ist das ein fairer W\"urfel?
+  \part Speicher die berechneten Wahrscheinlichkeiten $P(x_i)$ f\"ur das Auftreten der 
+  gew\"urfelten Zahlen in einem Vektor und benutze die \code{bar} Funktion,
+  um diese Wahrscheinlichkeiten als Funktion der Augenzahl zu plotten.
+  \part Erstelle in einem weiterem Plot ein entsprechendes normiertes Histogramm
+  mit der \code{hist} Funktion.
+  \part \extra Wie k\"onnte man einen gezinkten W\"urfel simulieren, bei dem die sechs
+  dreimal so h\"aufig wie die anderen Zahlen gew\"urfelt wird?\\
+  Fertige von diesem W\"urfel ein Histogram aus 10000 W\"urfen an.
+\end{parts}
+
+
+\continue
+\question \qt{Wahrscheinlichkeiten eines W\"urfels II}
+Wir werten nun das Verhalten mehrerer W\"urfel aus.
+\begin{parts}
+  \part Simuliere 20 W\"urfel, von denen jeder 100 mal geworfen wird 
+  (jeder W\"urfel wird mit dem gleichen Zufallsgenerator simuliert).
+  \part Berechne aus diesem Datensatz f\"ur jeden W\"urfel ein normiertes Histogramm.
+  \part Bestimme den Mittelwert und die Standardabweichung f\"ur jede
+  Augenzahl gemittelt \"uber die W\"urfel.
+  \part Stelle das Ergebnis mit einem S\"aulenplot mit Fehlerbalken dar
+  (\code{bar} mit \code{errorbar} Funktionen).
+\end{parts}
+
+
+\question \qt{Wahrscheinlichkeiten der Normalverteilung}
+Mit den folgenden Aufgaben wollen wir bestimmen, welcher Anteil eines
+normalverteilten Datensatzes in bestimmten Grenzen symmetrisch um den
+Mittelwert enthalten ist.
+\begin{parts}
+  \part Erzeuge einen Datensatz $X = (x_1, x_2, ... x_n)$ aus
+  $n=10000$ normalverteilten Zufallszahlen mit Mittelwert $\mu=0$ und
+  Standardabweichung $\sigma=1$.
+  \part \label{onesigma} Wieviele dieser Daten sind maximal eine Standardabweichung vom Mittelwert entfernt?\\
+  D.h. wieviele Datenwerte $x_i$ haben den Wert $-\sigma < x_i < +\sigma$?\\
+  Wie gro{\ss} ist also die Wahrscheinlichkeit $P_{\pm\sigma}$ einen
+  Wert in diesem Interval zu erhalten?
+  \part \label{probintegral} Berechne numerisch diese
+  Wahrscheinlichkeit aus dem entsprechenden Integral
+  \[ P_{\pm\sigma}=\int_{x=\mu-\sigma}^{x=\mu+\sigma} p_g(x) \, dx \]
+  \"uber die Normalverteilung
+  \[ p_g(x) =
+  \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \; . \]
+  \"Uberpr\"ufe zuerst, ob tats\"achlich
+  \[ \int_{-\infty}^{+\infty} p_g(x) \, dx = 1 \; . \]
+  Warum muss das so sein?
+  \part Welcher Anteil der Daten ist in den Intervallen $\pm2\sigma$ sowie $\pm3\sigma$
+  enthalten?
+  \part \label{givenfraction} Finde heraus in welchem Interval symmetrisch um den Mittelwert
+  50\,\%, 90\,\%, 95\,\% bzw. 99\,\% der Daten enhalten sind.
+  \part Was passiert mit der Wahrscheinlichkeit eine Zahl in einem bestimmten Interval
+  zu ziehen, wenn dieses Intervall immer kleiner wird?\\
+  Schreibe ein Programm, das dies illustriert.\\
+  Wie gro{\ss} ist die Wahrscheinlichkeit $P(x_i=0.1234)$?
+  \part \extra Modifiziere den Code der Teilaufgaben \pref{onesigma}
+  -- \pref{givenfraction} so, dass er f\"ur Datens\"atze mit
+  beliebigen Mittelwerten und Standardabweichungen funktioniert.\\
+  Teste den Code mit entsprechenden Zufallszahlen.\\
+  Wie bekommt man mit \code{randn} Zufallszahlen mit beliebiger
+  Standardabweichung und Mittelwerten?
+\end{parts}
+
+
+\end{questions}
+
+\end{document}
\ No newline at end of file
diff --git a/statistics/exercises/descriptivestatistics-02.tex b/statistics/exercises/descriptivestatistics-02.tex
new file mode 100644
index 0000000..3f92aa5
--- /dev/null
+++ b/statistics/exercises/descriptivestatistics-02.tex
@@ -0,0 +1,164 @@
+\documentclass[12pt,a4paper,pdftex]{exam}
+
+\usepackage[german]{babel}
+\usepackage{natbib}
+\usepackage{graphicx}
+\usepackage[small]{caption}
+\usepackage{sidecap}
+\usepackage{pslatex}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\setlength{\marginparwidth}{2cm}
+\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
+
+%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
+\pagestyle{headandfoot}
+\header{{\bfseries\large \"Ubung 2}}{{\bfseries\large Deskriptive Statistik}}{{\bfseries\large 19. Oktober, 2015}}
+\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
+jan.grewe@uni-tuebingen.de}
+\runningfooter{}{\thepage}{}
+
+\setlength{\baselineskip}{15pt}
+\setlength{\parindent}{0.0cm}
+\setlength{\parskip}{0.3cm}
+\renewcommand{\baselinestretch}{1.15}
+
+\newcommand{\qt}[1]{\textbf{#1}\\}
+\newcommand{\pref}[1]{(\ref{#1})}
+\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
+\newcommand{\code}[1]{\texttt{#1}}
+
+\newcommand{\continue}{\ifprintanswers%
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\continuepage}{\ifprintanswers%
+\newpage
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\newsolutionpage}{\ifprintanswers%
+\newpage%
+\else
+\fi}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+
+\vspace*{-6.5ex}
+\begin{center}
+\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex]
+{\large Jan Grewe, Jan Benda}\\[-3ex]
+Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
+\end{center}
+
+% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
+% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
+% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
+% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
+% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
+% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
+% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
+% (z.B. variablen\_datentypen\_mueller.m).
+
+
+\begin{itemize}
+\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie
+auch wirklich das macht, was sie machen soll! Teste dies mit kleinen
+Beispielen direkt in der Kommandozeile.
+\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in
+sinnvolle kleine Funktionen herunterzubrechen.
+\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
+lohnt es sich eine Funktion daraus zu schreiben!
+\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen
+Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles
+stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute
+Statistik zu bekommen.
+\item Benutze die Hilfsfunktion von matlab und das Internet, um
+herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden
+sind und was f\"ur M\"oglichkeiten sie bieten.
+\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten!
+\end{itemize}
+
+\begin{questions}
+
+\question \qt{Zentraler Grenzwertsatz}
+Der Zentrale Grenzwertsatz besagt, dass die Summe von unabh\"angigen
+und identisch verteilten (i.i.d. = independent and identically
+distributed) Zufallsvariablen gegen die Normalverteilung konvergiert.
+
+Den Zentralen Grenzwertsatz wollen wir uns im Folgenden veranschaulichen.
+\begin{parts}
+  \part Versuche dir klar zu machen, was der Zentrale Grenzwertsatz
+  bedeutet, und wie du vorgehen k\"onntest ein Programm zu
+  schreiben, das den Grenzwertsatz illustriert.
+  \part Erzeuge 10000 zwischen 0 und 1 gleichverteilte Zufallszahlen
+  (Funktion \code{rand}).
+  \part Plotte deren Wahrscheinlichkeitsdichte (normiertes Histogram).
+  \part Erzeuge weitere 10000 gleichverteilte Zufallszahlen und
+  addiere diese zu den bereits vorhandenen auf.
+  \part Plotte die Wahrscheinlichkeitsdichte der aufsummierten
+  Zufallszahlen.
+  \part Wiederhole Schritt (d) und (e) viele Male.
+  \part Vergleiche in einer Grafik die Wahrscheinlichkeitsdichte der
+  aufsummierten Zufallszahlen mit der Gaussfunktion
+  \[ p_g(x) =
+  \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]
+  mit dem Mittelwert $\mu$ und der Standardabweichung $\sigma$ der
+  aufsummierten Zufallszahlen.
+  \part Wie \"andert sich der Mittelwert und die
+  Standardabweichung/Varianz
+  der aufsummierten Zufallszahlen?\\
+  Wie h\"angen diese mit den Werten der urspr\"unglichen Verteilung
+  zusammen?
+  \part \extra \"Uberpr\"ufe den Grenzwertsatz in gleicher Weise mit exponentiell
+  verteilten Zufallszahlen (Funktion \code{rande}).
+\end{parts}
+
+
+\question \qt{Random Walk}
+Im folgenden wollen wir einige Eigenschaften des Random Walks bestimmen.
+\begin{parts}
+  \part Schreibe eine Funktion, die einen einzelnen Random Walk mit
+  Startwert 0 f\"ur $n$ Schritte und Wahrscheinlichkeit $p$ f\"ur
+  einen positiven Schritt als Vektor zur\"uckgibt.
+  \part Visualisiere jeweils 10 Random Walks mit $p=0.5$ zusammen in einem Plot
+  f\"ur $n=100$, $n=1000$ und $n=10000$ (drei Plots).\\
+  Sch\"atze aus den Abbildungen ab, wie sich der Mittelwert und die Standardabweichung
+  des Random Walks mit der Zeit (Schritte) sich entwickelt.
+  \part \"Uberpr\"uefe deine Hypothese zum Mittelwert und zur
+  Standardabweichung, indem du von $m$ Random Walks ($m \ge 10$) f\"ur
+  jeden z.B. zehnten Schritt den Mittelwert und die Standardabweichung
+  \"uber die Positionen der $m$ Random Walks berechnest.\\
+  Wie h\"angt also die Standardabweichung von der Anzahl der Schritte
+  ab? Wie entwickelt sich die Standardabweichung f\"ur eine sehr
+  gro{\ss}e Anzahl von Schritten?
+  \part \extra Erstelle eine Grafik, die die Verteilung der Position eines Random Walkers
+  zu drei verschiedenen Zeitpunkten zeigt.
+\end{parts}
+
+
+\question \qt{\extra 2D Random Walk}
+Bisher hat sich unser Random Walker nur in einer Dimension bewegt 
+(nur vorw\"arts oder r\"uckw\"arts). Er kann aber auch in mehreren Dimensionen laufen!\\
+In zwei Dimensionen wird dazu in jedem Schritt eine weitere
+Zufallszahl gezogen, die bestimmt ob er einen Schritt nach links oder
+rechts gemacht hat. Die Bewegung nach vorne/hinten bzw. links/rechts
+sind unabh\"angig voneinander.
+\begin{parts}
+  \part Wie kann unter Verwendung unserer Funktion f\"ur den
+  eindimensionalen Random Walk ein zweidimensionaler Random Walk
+  simuliert werden?
+  \part Erstelle h\"ubsche Bilder, die zweidimensionalen Random
+  Walks verschiedener L\"ange (bis zu mindestens $n=1000000$) illustrieren.
+  \part Animationen sind auch sch\"on! z.B. mit dem \code{pause} Befehl.
+  \part Anstatt einfach den Weg des Random Walks zu zeichnen, kann man
+  sich auch merken, wie oft er an jeder Stelle vorbeigekommen ist und
+  mit einem Farbcode plotten.
+\end{parts}
+
+\end{questions}
+
+\end{document}
\ No newline at end of file
diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex
index 3ae32ac..989dc0f 100644
--- a/statistics/lecture/descriptivestatistics.tex
+++ b/statistics/lecture/descriptivestatistics.tex
@@ -74,8 +74,8 @@
 \newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
   {\medskip}
 
-%\newcommand{\showlisting}{yes}
-\newcommand{\showlisting}{no}
+\newcommand{\showlisting}{yes}
+%\newcommand{\showlisting}{no}
 \newcounter{theexercise} 
 \setcounter{theexercise}{1}
 \newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}

From 1e3b02b9a1bd2d17fede25c756dcbfca5134ff2e Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Mon, 19 Oct 2015 18:53:52 +0200
Subject: [PATCH 2/4] Few updates on todays lecture

---
 statistics/code/gaussianbins.m               |  7 +++---
 statistics/code/gaussianpdf.m                | 26 +++++++++++++-------
 statistics/code/mymedian.m                   | 10 ++++----
 statistics/code/quartiles.m                  |  9 ++++---
 statistics/lecture/Makefile                  |  7 +++++-
 statistics/lecture/descriptivestatistics.tex | 15 ++++++-----
 6 files changed, 44 insertions(+), 30 deletions(-)

diff --git a/statistics/code/gaussianbins.m b/statistics/code/gaussianbins.m
index 5f3cc6d..effe1fd 100644
--- a/statistics/code/gaussianbins.m
+++ b/statistics/code/gaussianbins.m
@@ -1,6 +1,6 @@
-x = randn( 100, 1 );
-bins1 = -4:2:4;
-bins2 = -4:0.5:4;
+x = randn( 100, 1 );   % generate some data
+bins1 = -4:2:4;        % large bins
+bins2 = -4:0.5:4;      % small bins
 subplot( 1, 2, 1 );
 hold on;
 hist( x, bins1 );
@@ -10,6 +10,7 @@ ylabel('Frequeny')
 hold off;
 subplot( 1, 2, 2 );
 hold on;
+% normalize to the rigtht bin size:
 hist( x, bins1, 1.0/(bins1(2)-bins1(1)) );
 hist( x, bins2, 1.0/(bins2(2)-bins2(1)) );
 xlabel('x')
diff --git a/statistics/code/gaussianpdf.m b/statistics/code/gaussianpdf.m
index 6c33dc6..259d6c9 100644
--- a/statistics/code/gaussianpdf.m
+++ b/statistics/code/gaussianpdf.m
@@ -1,22 +1,30 @@
 % plot Gaussian pdf:
-dx=0.1
+dx=0.1;
 x = [-4.0:dx:4.0];
 p = exp(-0.5*x.^2)/sqrt(2.0*pi);
 hold on
-plot(x,p, 'linewidth', 10 )
+plot(x, p, 'linewidth', 10)
+% show area of integral:
+area(x((x>=x1)&(x<=x2)), p((x>=x1)&(x<=x2)), 'FaceColor', 'r' )
+hold off
 
 % compute integral between x1 and x2:
-x1=1.0
-x2=2.0
-P = sum(p((x>=x1)&(x<x2)))*dx
+x1=1.0;
+x2=2.0;
+P = sum(p((x>=x1)&(x<x2)))*dx;
+disp( [ 'The integral between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(P, 3) ] );
 
 % draw random numbers:
-r = randn( 10000, 1 );
-hist(r,x,1.0/dx)
+%r = randn( 10000, 1 );
+%hist(r,x,1.0/dx)
 
 % check P:
-Pr = sum((r>=x1)&(r<x2))/length(r)
+Pr = sum((r>=x1)&(r<x2))/length(r);
+disp( [ 'The probability of getting a number between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(Pr, 3) ] );
 
-hold off
+% infinite integral:
+P = sum(p)*dx;
+disp( [ 'The integral between -infinity and +infinity is ', num2str(P, 3) ] );
+disp( [ 'I.e. the probability to get any number is ', num2str(P, 3) ] );
 
 
diff --git a/statistics/code/mymedian.m b/statistics/code/mymedian.m
index d2af4d0..1f88193 100644
--- a/statistics/code/mymedian.m
+++ b/statistics/code/mymedian.m
@@ -1,13 +1,13 @@
 function m = mymedian( x )
 % returns the median of the vector x
   xs = sort( x );
-  if ( length( xs ) == 0 )
+  if ( length( xs ) == 0 )                 % empty input vector
     m = NaN;
-  elseif ( rem( length( xs ), 2 ) == 0 )
+  elseif ( rem( length( xs ), 2 ) == 0 )   % even number of data values
     index = length( xs )/2;
-    m = (xs( index ) + xs( index+1 ))/2;
-  else
-    index = (length( xs ) + 1)/2;
+    m = (xs( index ) + xs( index+1 ))/2;   % average the two central elements
+  else                                     % odd number of data values
+    index = (length( xs ) + 1)/2;          % take the middle element
     m = xs( index );
   end
 end
diff --git a/statistics/code/quartiles.m b/statistics/code/quartiles.m
index 3f9ebe5..8123733 100644
--- a/statistics/code/quartiles.m
+++ b/statistics/code/quartiles.m
@@ -1,13 +1,14 @@
 function q = quartiles( x )
-  % returns a vector with the first, second, and third quartile of the vector x
+  % returns a vector with the first, second, and third quartile
+  % of the vector x
   xs = sort( x );
-  if ( length( xs ) == 0 )
+  if ( length( xs ) == 0 )                            % no data
     q = [];
-  elseif ( rem( length( xs ), 2 ) == 0 )
+  elseif ( rem( length( xs ), 2 ) == 0 )              % even number of data
     index = length( xs )/2;
     m = (xs( index ) + xs( index+1 ))/2;
     q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ];
-  else
+  else                                                % odd number of data
     index = (length( xs ) + 1)/2;
     m = xs( index );
     q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ];
diff --git a/statistics/lecture/Makefile b/statistics/lecture/Makefile
index b68ae15..59963db 100644
--- a/statistics/lecture/Makefile
+++ b/statistics/lecture/Makefile
@@ -1,11 +1,16 @@
 TEXFILES=$(wildcard *.tex)
 PDFFILES=$(TEXFILES:.tex=.pdf)
+PYFILES=$(wildcard *.py)
+PYPDFFILES=$(PYFILES:.py=.pdf)
 
-pdf : $(PDFFILES)
+pdf : $(PDFFILES) $(PYPDFFILES)
 
 $(PDFFILES) : %.pdf : %.tex
 	pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
 
+$(PYPDFFILES) : %.pdf : %.py
+	python $<
+
 clean :
 	rm -f *~ $(TEXFILES:.tex=.aux) $(TEXFILES:.tex=.log) $(TEXFILES:.tex=.out) $(TEXFILES:.tex=.nav) $(TEXFILES:.tex=.snm) $(TEXFILES:.tex=.toc) $(TEXFILES:.tex=.vrb)
 
diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex
index 989dc0f..bd20f5a 100644
--- a/statistics/lecture/descriptivestatistics.tex
+++ b/statistics/lecture/descriptivestatistics.tex
@@ -154,7 +154,7 @@
     below. In particular the script should test data vectors of
     different length.}  {Schreibe ein Skript, das testet ob die
     \code{mymedian} Funktion wirklich die Zahl zur\"uckgibt, \"uber
-    der genausoviele Datenwerte liegen wie darunter. Das Skript sollte
+    der genauso viele Datenwerte liegen wie darunter. Das Skript sollte
     insbesondere verschieden lange Datenvektoren testen.}
 \end{exercise}
 
@@ -246,7 +246,7 @@ $A$ des Histogramms ist also
 \[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \]
 und das normierte Histogramm hat die H\"ohe
 \[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \]
-Es muss also nicht nur durch die Summe, sondern auch durch die Breite der Klassen $\Delta x$
+Es muss also nicht nur durch die Summe, sondern auch durch die Breite $\Delta x$ der Klassen
 geteilt werden.
 
 $p(x_i)$ kann keine Wahrscheinlichkeit sein, da $p(x_i)$ nun eine
@@ -258,14 +258,14 @@ spricht von einer Wahrscheinlichkeitsdichte.
   \caption{\label{pdfprobabilitiesfig} Wahrscheinlichkeiten bei
   einer Wahrscheinlichkeitsdichtefunktion.}
 \end{figure}
-  
-\begin{exercise}
+
+\begin{exercise}[gaussianpdf.m]
   \tr{Plot the Gaussian probability density}{Plotte die Gauss'sche Wahrscheinlichkeitsdichte }
-  \[ p_g(x) = 1/\sqrt{2\pi\sigma^2}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
+  \[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
   \tr{What does it mean?}{Was bedeutet die folgende Wahrscheinlichkeit?}
-  \[ P(x_1 < x < x2) = \int_{x_1}^{x_2} p(x) \, dx \]
+  \[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \]
   \tr{How large is}{Wie gro{\ss} ist}
-  \[ \int_{-\infty}^{+\infty} p(x) \, dx \; ?\]
+  \[ \int\limits_{-\infty}^{+\infty} p(x) \, dx \; ?\]
   \tr{Why?}{Warum?}
 \end{exercise}
 
@@ -392,7 +392,6 @@ spricht von einer Wahrscheinlichkeitsdichte.
 
 \end{document}
 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Statistics}
 What is "a statistic"? % dt. Sch\"atzfunktion

From 2787e55094949b099fec2cf7a967ef27d5317cac Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Tue, 20 Oct 2015 01:43:18 +0200
Subject: [PATCH 3/4] New bootstrap exercises

---
 statistics/code/bootstrapsem.m                |    23 +
 statistics/code/bootstraptymus.m              |    24 +
 statistics/code/boxwhisker.m                  |     3 +
 statistics/code/correlations.m                |    14 +
 statistics/code/tdistribution.m               |    32 +
 statistics/exercises/bootstrap-01.tex         |   132 +
 statistics/exercises/thymusglandweights.dat   | 10000 ++++++++++++++++
 statistics/lecture/boxwhisker.py              |    47 +
 statistics/lecture/boxwhiskerdata.npy         |   Bin 0 -> 3280 bytes
 statistics/lecture/descriptivestatistics.tex  |    85 +
 .../figures/2012-10-29_16-26-05_771.jpg       |   Bin 0 -> 740827 bytes
 .../figures/2012-10-29_16-29-35_312.jpg       |   Bin 0 -> 395085 bytes
 .../figures/2012-10-29_16-41-39_523.jpg       |   Bin 0 -> 472511 bytes
 statistics/lecture/figures/srs1.png           |   Bin 0 -> 60051 bytes
 statistics/lecture/figures/srs2.png           |   Bin 0 -> 56148 bytes
 statistics/lecture/figures/srs3.png           |   Bin 0 -> 74342 bytes
 statistics/material/Hesterberg2003.pdf        |   Bin 0 -> 589657 bytes
 17 files changed, 10360 insertions(+)
 create mode 100644 statistics/code/bootstrapsem.m
 create mode 100644 statistics/code/bootstraptymus.m
 create mode 100644 statistics/code/boxwhisker.m
 create mode 100644 statistics/code/correlations.m
 create mode 100644 statistics/code/tdistribution.m
 create mode 100644 statistics/exercises/bootstrap-01.tex
 create mode 100644 statistics/exercises/thymusglandweights.dat
 create mode 100644 statistics/lecture/boxwhisker.py
 create mode 100644 statistics/lecture/boxwhiskerdata.npy
 create mode 100755 statistics/lecture/figures/2012-10-29_16-26-05_771.jpg
 create mode 100755 statistics/lecture/figures/2012-10-29_16-29-35_312.jpg
 create mode 100755 statistics/lecture/figures/2012-10-29_16-41-39_523.jpg
 create mode 100644 statistics/lecture/figures/srs1.png
 create mode 100644 statistics/lecture/figures/srs2.png
 create mode 100644 statistics/lecture/figures/srs3.png
 create mode 100644 statistics/material/Hesterberg2003.pdf

diff --git a/statistics/code/bootstrapsem.m b/statistics/code/bootstrapsem.m
new file mode 100644
index 0000000..b5a7cbe
--- /dev/null
+++ b/statistics/code/bootstrapsem.m
@@ -0,0 +1,23 @@
+nsamples = 1000
+resample = 500
+
+  x = randn( nsamples, 1 );
+sem = std(x)/sqrt(nsamples);
+
+mu = zeros( resample, 1 );
+for i = 1:resample
+  % resample:
+  xr = x(randi(nsamples, nsamples, 1));
+  % compute statistics on sample:
+  mu(i) = mean(xr);
+end
+bootsem = std( mu );
+
+hold on
+hist( x, 20 );
+hist( mu, 20 );
+hold off
+
+disp(['bootstrap standard error: ', num2str(bootsem)]);
+disp(['standard error: ', num2str(sem)]);
+
diff --git a/statistics/code/bootstraptymus.m b/statistics/code/bootstraptymus.m
new file mode 100644
index 0000000..e92a5b7
--- /dev/null
+++ b/statistics/code/bootstraptymus.m
@@ -0,0 +1,24 @@
+resample = 500
+
+load( 'thymusglandweights.dat' );
+x = thymusglandweights;
+nsamples = length( x );
+sem = std(x)/sqrt(nsamples);
+
+mu = zeros( resample, 1 );
+for i = 1:resample
+  % resample:
+  xr = x(randi(nsamples, nsamples, 1));
+  % compute statistics on sample:
+  mu(i) = mean(xr);
+end
+bootsem = std( mu );
+
+hold on
+hist( x, 20 );
+hist( mu, 20 );
+hold off
+
+disp(['bootstrap standard error: ', num2str(bootsem)]);
+disp(['standard error: ', num2str(sem)]);
+
diff --git a/statistics/code/boxwhisker.m b/statistics/code/boxwhisker.m
new file mode 100644
index 0000000..65e8f24
--- /dev/null
+++ b/statistics/code/boxwhisker.m
@@ -0,0 +1,3 @@
+x = randn(40, 10);
+boxplot(x, 'whisker', 100.0 );
+
diff --git a/statistics/code/correlations.m b/statistics/code/correlations.m
new file mode 100644
index 0000000..7816457
--- /dev/null
+++ b/statistics/code/correlations.m
@@ -0,0 +1,14 @@
+n = 1000
+x = randn( n, 1 );
+y = randn( n, 1 ) + 0.2*x;
+r = corr(x,y)
+
+  nsamples = 500;
+  rs = zeros( nsamples, 1 );
+for i = 1:nsamples
+	  xs = x(randi(n,n,1));
+ys = x(randi(n,n,1));
+	  rs(i) = corr(xs,ys);
+end
+
+hist( rs, 20 )
diff --git a/statistics/code/tdistribution.m b/statistics/code/tdistribution.m
new file mode 100644
index 0000000..9d8920b
--- /dev/null
+++ b/statistics/code/tdistribution.m
@@ -0,0 +1,32 @@
+n = 100000
+x=randn(n, 1);
+
+nsamples = 3;
+nmeans = 10000;
+means = zeros( nmeans, 1 );
+sdevs = zeros( nmeans, 1 );
+students = zeros( nmeans, 1 );
+for i=1:nmeans
+  sample = x(randi(n, nsamples, 1));
+  means(i) = mean(sample);
+  sdevs(i) = std(sample);
+  students(i) = mean(sample)/std(sample)*sqrt(nsamples);
+end
+sdev = 1.0
+msdev = std(means)
+
+%  scatter( means, sdevs )
+
+hold on;
+dxg=0.01;
+xmax = 10.0
+xmin = -xmax
+xg = [xmin:dxg:xmax];
+pg = exp(-0.5*(xg/sdev).^2)/sqrt(2.0*pi)/sdev;
+hold on
+plot(xg, pg, 'r', 'linewidth', 4)
+
+bins = xmin:0.1:xmax;
+hist(means, bins, 1.0/(bins(2)-bins(1)) );
+hold off;
+
diff --git a/statistics/exercises/bootstrap-01.tex b/statistics/exercises/bootstrap-01.tex
new file mode 100644
index 0000000..a5860e1
--- /dev/null
+++ b/statistics/exercises/bootstrap-01.tex
@@ -0,0 +1,132 @@
+\documentclass[12pt,a4paper,pdftex]{exam}
+
+\usepackage[german]{babel}
+\usepackage{natbib}
+\usepackage{graphicx}
+\usepackage[small]{caption}
+\usepackage{sidecap}
+\usepackage{pslatex}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\setlength{\marginparwidth}{2cm}
+\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
+
+%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
+\pagestyle{headandfoot}
+\header{{\bfseries\large \"Ubung 1}}{{\bfseries\large Bootstrap}}{{\bfseries\large 21. Oktober, 2015}}
+\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
+jan.benda@uni-tuebingen.de}
+\runningfooter{}{\thepage}{}
+
+\setlength{\baselineskip}{15pt}
+\setlength{\parindent}{0.0cm}
+\setlength{\parskip}{0.3cm}
+\renewcommand{\baselinestretch}{1.15}
+
+\newcommand{\qt}[1]{\textbf{#1}\\}
+\newcommand{\pref}[1]{(\ref{#1})}
+\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
+\newcommand{\code}[1]{\texttt{#1}}
+
+\newcommand{\continue}{\ifprintanswers%
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\continuepage}{\ifprintanswers%
+\newpage
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\newsolutionpage}{\ifprintanswers%
+\newpage%
+\else
+\fi}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+
+\vspace*{-6.5ex}
+\begin{center}
+\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex]
+{\large Jan Grewe, Jan Benda}\\[-3ex]
+Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
+\end{center}
+
+\begin{itemize}
+\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie
+auch wirklich das macht, was sie machen soll! Teste dies mit kleinen
+Beispielen direkt in der Kommandozeile.
+\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in
+sinnvolle kleine Funktionen herunterzubrechen.
+\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
+lohnt es sich eine Funktion daraus zu schreiben!
+\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen
+Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles
+stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute
+Statistik zu bekommen.
+\item Benutze die Hilfsfunktion von matlab und das Internet, um
+herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden
+sind und was f\"ur M\"oglichkeiten sie bieten.
+\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten!
+\end{itemize}
+
+
+\begin{questions}
+
+\question \qt{Bootstrap des Standardfehlers}
+\begin{parts}
+  \part Lade von Ilias die Datei \code{thymusglandweights.dat} herunter.
+  Darin befindet sich ein Datensatz vom Gewicht der Thymus Dr\"use in 14-Tage alten
+  H\"uhnerembryos in mg.
+  \part Lade diese Daten in Matlab (\code{load} Funktion).
+  \part Bestimme Histogramm, Mittelwert und Standardfehler aus den ersten 80 Datenpunkten.
+  \part Bestimme den Standardfehler aus den ersten 80 Datenpunkten durch 500 Mal Bootstrappen.
+  \part Bestimme das 95\,\% Konfidenzintervall f\"ur den Mittelwert
+  aus der Bootstrap Verteilung (\code{quantile()} Funktion) --- also
+  das Interval innerhalb dessen mit 95\,\% Wahrscheinlichkeit der
+  wahre Mittelwert liegen wird.
+  \part Benutze den ganzen Datensatz und die Bootstrapping Technik, um die Abh\"angigkeit
+  des Standardfehlers von der Stichprobengr\"o{\ss}e zu bestimmen.
+  \part Vergleiche mit der bekannten Formel f\"ur den Standardfehler $\sigma/\sqrt{n}$.
+\end{parts}
+
+
+\continue
+\question \qt{Student t-Verteilung}
+\begin{parts}
+\part Erzeuge 100000 normalverteilte Zufallszahlen.
+\part Ziehe daraus 1000 Stichproben vom Umfang $m$ (3, 5, 10, 50).
+\part Berechne den Mittelwert $\bar x$ der Stichproben und plotte die Wahrscheinlichkeitsdichte
+dieser Mittelwerte.
+\part Vergleiche diese Wahrscheinlichkeitsdichte mit der Gausskurve.
+\part Berechne ausserdem die Gr\"o{\ss}e $t=\bar x/(\sigma_x/\sqrt{m}$
+(Standardabweichung $\sigma_x$) und vergleiche diese mit der Normalverteilung mit Standardabweichung Eins. Ist $t$ normalverteilt, bzw. unter welchen Bedingungen ist $t$ Normalverteilt?
+\end{parts}
+
+
+\question \qt{Korrelationen}
+\begin{parts}
+\part Erzeuge 1000 korrelierte Zufallszahlen $x$, $y$ durch
+\begin{verbatim}
+n = 1000
+a = 0.2;
+x = randn(n, 1);
+y = randn(n, 1) + a*x;
+\end{verbatim}
+\part Erstelle einen Scatterplot der beiden Variablen.
+\part Warum ist $y$ mit $x$ korreliert?
+\part Berechne den Korrelationskoeffizienten zwischen $x$ und $y$.
+\part Was m\"usste man tun, um die Korrelationen zwischen den $x$-$y$
+Paaren zu zerst\"oren?
+\part Mach genau dies 1000 mal und berechne jedes Mal den Korrelationskoeffizienten.
+\part Bestimme die Wahrscheinlichkeitsdichte dieser Korrelationskoeffizienten.
+\part Ist die Korrelation der urspr\"unglichen Daten signifikant?
+\part Variiere den Parameter $a$ und \"uberpr\"ufe auf gleiche Weise die Signifikanz.
+\end{parts}
+
+
+\end{questions}
+
+\end{document}
\ No newline at end of file
diff --git a/statistics/exercises/thymusglandweights.dat b/statistics/exercises/thymusglandweights.dat
new file mode 100644
index 0000000..7427de9
--- /dev/null
+++ b/statistics/exercises/thymusglandweights.dat
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diff --git a/statistics/lecture/boxwhisker.py b/statistics/lecture/boxwhisker.py
new file mode 100644
index 0000000..1209f7e
--- /dev/null
+++ b/statistics/lecture/boxwhisker.py
@@ -0,0 +1,47 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+#x = np.random.randn( 40, 10 )
+#np.save('boxwhiskerdata', x )
+x = np.load('boxwhiskerdata.npy')
+
+plt.xkcd()
+fig = plt.figure( figsize=(6,4) )
+ax = fig.add_subplot( 1, 1, 1 )
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.set_xlabel('Experiment')
+ax.set_ylabel('x')
+ax.set_ylim( -4.0, 4.0)
+ax.annotate('Median',
+            xy=(3.9, 0.1), xycoords='data',
+            xytext=(3.5, -2.5), textcoords='data', ha='right',
+            arrowprops=dict(arrowstyle="->", relpos=(0.8,1.0),
+            connectionstyle="angle3,angleA=-110,angleB=60") )
+ax.annotate('1. quartile',
+            xy=(5.8, -0.7), xycoords='data',
+            xytext=(5.5, -3.5), textcoords='data', ha='right',
+            arrowprops=dict(arrowstyle="->", relpos=(0.5,1.0),
+            connectionstyle="angle3,angleA=30,angleB=70") )
+ax.annotate('3. quartile',
+            xy=(6.1, 0.6), xycoords='data',
+            xytext=(6.5, 3.0), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(0.0,0.0),
+            connectionstyle="angle3,angleA=30,angleB=70") )
+ax.annotate('minimum',
+            xy=(6.1, -2.3), xycoords='data',
+            xytext=(7.2, -3.3), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
+            connectionstyle="angle3,angleA=10,angleB=100") )
+ax.annotate('maximum',
+            xy=(5.9, 2.8), xycoords='data',
+            xytext=(4.9, 3.5), textcoords='data', ha='right',
+            arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
+            connectionstyle="angle3,angleA=0,angleB=120") )
+ax.boxplot( x, whis=100.0 )
+plt.tight_layout()
+plt.savefig('boxwhisker.pdf')
+plt.show()
+
diff --git a/statistics/lecture/boxwhiskerdata.npy b/statistics/lecture/boxwhiskerdata.npy
new file mode 100644
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diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex
index bd20f5a..21e7493 100644
--- a/statistics/lecture/descriptivestatistics.tex
+++ b/statistics/lecture/descriptivestatistics.tex
@@ -82,6 +82,8 @@
   \arabic{theexercise}:} \stepcounter{theexercise}\newline \newcommand{\exercisesource}{#1}}%
   {\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\equal{\showlisting}{yes}}{\medskip\lstinputlisting{\exercisesource}}{}}\medskip}
 
+\graphicspath{{figures/}}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -269,6 +271,24 @@ spricht von einer Wahrscheinlichkeitsdichte.
   \tr{Why?}{Warum?}
 \end{exercise}
 
+\begin{exercise}[boxwhisker.m]
+  \tr{Generate eine $40 \times 10$ matrix of random numbers and
+    illustrate their distribution in a box-whicker plot
+    (\code{boxplot()} function). How to interpret the plot?}
+  {Erzeuge ein $40 \times 10$ Matrix
+    von Zufallszahlen und illustriere ihre Verteilungen in einem
+    Box-Whisker Plot (\code{boxplot()} Funktion, lies die Hilfe!). Wie ist der
+    Box-Whisker Plot zu interpretieren? Was hat es mit den Ausreissern auf sich?
+    Wie kann man erreichen, dass die Whisker den kleinsten und den gr\"o{\ss}ten
+    Datenwert anzeigen? Warum sind die unterschiedlichen Box-Whiskers nicht alle gleich,
+    obwohl sie aus der selben Verteilung gezogen worden sind?}
+\end{exercise}
+
+\begin{figure}[t]
+  \includegraphics[width=1\textwidth]{boxwhisker}
+  \caption{\label{boxwhiskerfig} Box-whisker plots illustrate distributions.}
+\end{figure}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Data types}
@@ -390,6 +410,71 @@ spricht von einer Wahrscheinlichkeitsdichte.
 \end{itemize}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\chapter{\tr{Bootstrap Methods}{Bootstrap Methoden}}
+
+Beim Bootstrap erzeugt man sich die Verteilung von Statistiken durch Resampling
+aus der Stichprobe. Das hat mehrere Vorteile:
+\begin{itemize}
+\item Weniger Annahmen (z.B. muss eine Stichprobe nicht Normalverteilt sein).
+\item H\"ohere Genauigkeit als klassische Methoden.
+\item Allgemeing\"ultigkeit: Bootstrap Methoden sind sich sehr
+  \"ahnlich f\"ur viele verschiedene Statistiken und ben\"otigen nicht
+  f\"ur jede Statistik eine andere Formel.
+\end{itemize}
+
+\begin{figure}[t]
+  \includegraphics[width=0.8\textwidth]{2012-10-29_16-26-05_771}\\[2ex]
+  \includegraphics[width=0.8\textwidth]{2012-10-29_16-41-39_523}\\[2ex]
+  \includegraphics[width=0.8\textwidth]{2012-10-29_16-29-35_312}
+  \caption{\tr{Why can we only measure a sample of the
+      population?}{Warum k\"onnen wir nur eine Stichprobe der
+      Grundgesamtheit messen?}}
+\end{figure}
+
+\begin{figure}[t]
+  \includegraphics[height=0.2\textheight]{srs1}\\[2ex]
+  \includegraphics[height=0.2\textheight]{srs2}\\[2ex]
+  \includegraphics[height=0.2\textheight]{srs3}
+  \caption{Bootstrap der Stichprobenvertielung (a) Von der
+    Grundgesamtheit (population) mit unbekanntem Parameter
+    (z.B. Mittelwert $\mu$) zieht man Stichproben (SRS: simple random
+    samples).  Die Statistik (hier Bestimmung von $\bar x$) kann f\"ur
+    jede Stichprobe berechnet werden. Die erhaltenen Werte entstammen
+    der Stichprobenverteilung. Meisten wird aber nur eine Stichprobe
+    gezogen!  (b) Mit bestimmten Annahmen und Theorien kann man auf
+    die Stichprobenverteilung schlie{\ss}en ohne sie gemessen zu
+    haben.  (c) Alternativ k\"onnen aus der einen Stichprobe viele
+    Bootstrap-Stichproben generiert werden (resampling) und so
+    Eigenschaften der Stichprobenverteilung empirisch bestimmt
+    werden. Aus Hesterberg et al. 2003, Bootstrap Methods and
+    Permuation Tests}
+\end{figure}
+
+\section{Bootstrap des Standardfehlers}
+
+Beim Bootstrap erzeugen wir durch resampling neue Stichproben und
+benutzen diese um die Stichprobenverteilung einer Statistik zu
+berechnen. Die Bootstrap Stichproben haben jeweils den gleichen Umfang
+wie die urspr\"unglich gemessene Stichprobe und werden durch Ziehen
+mit Zur\"ucklegen gewonnen. Jeder Wert der urspr\"unglichen Stichprobe
+kann also einmal, mehrmals oder gar nicht in einer Bootstrap
+Stichprobe vorkommen.
+
+\begin{exercise}[bootstrapsem.m]
+  Ziehe 1000 normalverteilte Zufallszahlen und berechne deren Mittelwert,
+  Standardabweichung und Standardfehler ($\sigma/\sqrt{n}$).
+
+  Resample die Daten 1000 mal (Ziehen mit Zur\"ucklegen) und berechne jeweils
+  den Mittelwert.
+
+  Plotte ein Histogramm dieser Mittelwerte, sowie deren Mittelwert und
+  die Standardabweichung.
+
+  Was hat das mit dem Standardfehler zu tun?
+\end{exercise}
+
 \end{document}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/statistics/lecture/figures/2012-10-29_16-26-05_771.jpg b/statistics/lecture/figures/2012-10-29_16-26-05_771.jpg
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From 69b9afbc80a757e2363382b62608b9c24b1340c6 Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Tue, 20 Oct 2015 18:26:43 +0200
Subject: [PATCH 4/4] Added correlation plots

---
 statistics/code/diehistograms.m              |   6 +-
 statistics/lecture/correlation.py            |  34 ++++
 statistics/lecture/descriptivestatistics.tex | 165 ++++++++++++++++++-
 statistics/lecture/nonlincorrelation.py      |  42 +++++
 4 files changed, 242 insertions(+), 5 deletions(-)
 create mode 100644 statistics/lecture/correlation.py
 create mode 100644 statistics/lecture/nonlincorrelation.py

diff --git a/statistics/code/diehistograms.m b/statistics/code/diehistograms.m
index 8af7d68..d04ca98 100644
--- a/statistics/code/diehistograms.m
+++ b/statistics/code/diehistograms.m
@@ -3,16 +3,16 @@ nrolls = [ 20, 100, 1000 ];
 for i = [1:length(nrolls)]
   d = rollthedie( nrolls(i) );
   % plain hist:
-  % hist( d )
+  %hist( d )
 
   % check bin counts of plain hist:
   % h = hist( d )
 
   % force 6 bins:
-  % hist( d, 6 )
+  %hist( d, 6 )
 
   % set the right bin centers:
-  bins = 1:6;
+  %bins = 1:6;
   %hist( d, bins )
 
   % normalize histogram and compare to expectation:
diff --git a/statistics/lecture/correlation.py b/statistics/lecture/correlation.py
new file mode 100644
index 0000000..db317c7
--- /dev/null
+++ b/statistics/lecture/correlation.py
@@ -0,0 +1,34 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+plt.xkcd()
+fig = plt.figure( figsize=(6,5) )
+n = 200
+for k, r  in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
+    print r
+    x = np.random.randn( n )
+    y = r*x + np.sqrt(1.0-r*r)*np.random.randn( n )
+    ax = fig.add_subplot( 2, 2, k+1 )
+    ax.spines['right'].set_visible(False)
+    ax.spines['top'].set_visible(False)
+    ax.yaxis.set_ticks_position('left')
+    ax.xaxis.set_ticks_position('bottom')
+    ax.text( -2, 2.5, 'r=%.1f' % r )
+    if k == 0 :
+        ax.text( 2.8, -2, 'positively\ncorrelated', ha='right' )
+    elif k == 1 :
+        ax.text( 2.8, -2.5, 'weakly\ncorrelated', ha='right' )
+    elif k == 2 :
+        ax.text( 2.8, -2.5, 'not\ncorrelated', ha='right' )
+    elif k == 3 :
+        ax.text( -2.5, -2, 'negatively\ncorrelated', ha='left' )
+    ax.set_xlabel('x')
+    ax.set_ylabel('y')
+    ax.set_xlim( -3.0, 3.0)
+    ax.set_ylim( -3.0, 3.0)
+    ax.scatter( x, y )
+
+plt.tight_layout()
+plt.savefig('correlation.pdf')
+plt.show()
+
diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex
index 21e7493..ac99345 100644
--- a/statistics/lecture/descriptivestatistics.tex
+++ b/statistics/lecture/descriptivestatistics.tex
@@ -19,10 +19,141 @@
 \usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
 \setcounter{tocdepth}{1}
 
-%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[sf,bf,it,big,clearempty]{titlesec}
+\setcounter{secnumdepth}{-1}
+
+
+%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+ 
+
+%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \usepackage{graphicx}
 \usepackage{xcolor}
-\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
+\pagecolor{white}
+
+\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
+  \put(0,4){\line(1,0){170}}%
+  \multiput(0,2)(10,0){18}{\line(0,1){4}}%
+  \multiput(0,3)(1,0){170}{\line(0,1){2}}%
+  \put(0,0){\makebox(0,0){{\tiny 0}}}%
+  \put(10,0){\makebox(0,0){{\tiny 1}}}%
+  \put(20,0){\makebox(0,0){{\tiny 2}}}%
+  \put(30,0){\makebox(0,0){{\tiny 3}}}%
+  \put(40,0){\makebox(0,0){{\tiny 4}}}%
+  \put(50,0){\makebox(0,0){{\tiny 5}}}%
+  \put(60,0){\makebox(0,0){{\tiny 6}}}%
+  \put(70,0){\makebox(0,0){{\tiny 7}}}%
+  \put(80,0){\makebox(0,0){{\tiny 8}}}%
+  \put(90,0){\makebox(0,0){{\tiny 9}}}%
+  \put(100,0){\makebox(0,0){{\tiny 10}}}%
+  \put(110,0){\makebox(0,0){{\tiny 11}}}%
+  \put(120,0){\makebox(0,0){{\tiny 12}}}%
+  \put(130,0){\makebox(0,0){{\tiny 13}}}%
+  \put(140,0){\makebox(0,0){{\tiny 14}}}%
+  \put(150,0){\makebox(0,0){{\tiny 15}}}%
+  \put(160,0){\makebox(0,0){{\tiny 16}}}%
+  \put(170,0){\makebox(0,0){{\tiny 17}}}%
+  \end{picture}\par}
+
+% figures:
+\setlength{\fboxsep}{0pt}
+\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
+%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
+%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
+%\newcommand{\texpicture}[1]{}
+\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
+
+% maximum number of floats:
+\setcounter{topnumber}{2}
+\setcounter{bottomnumber}{0}
+\setcounter{totalnumber}{2}
+
+% float placement fractions:
+\renewcommand{\textfraction}{0.2}
+\renewcommand{\topfraction}{0.8}
+\renewcommand{\bottomfraction}{0.0}
+\renewcommand{\floatpagefraction}{0.5}
+
+% spacing for floats:
+\setlength{\floatsep}{12pt plus 2pt minus 2pt}
+\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
+\setlength{\intextsep}{12pt plus 2pt minus 2pt}
+
+% spacing for a floating page:
+\makeatletter
+  \setlength{\@fptop}{0pt}
+  \setlength{\@fpsep}{8pt plus 2.0fil}
+  \setlength{\@fpbot}{0pt plus 1.0fil}
+\makeatother
+
+% rules for floats:
+\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
+\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
+
+% captions:
+\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
+
+% put caption on separate float:
+\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
+
+% references to panels of a figure within the caption:
+\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
+% references to figures:
+\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
+\newcommand{\fref}[1]{\textup{\ref{#1}}}
+\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
+% references to figures in normal text:
+\newcommand{\fig}{Fig.}
+\newcommand{\Fig}{Figure}
+\newcommand{\figs}{Figs.}
+\newcommand{\Figs}{Figures}
+\newcommand{\figref}[1]{\fig~\fref{#1}}
+\newcommand{\Figref}[1]{\Fig~\fref{#1}}
+\newcommand{\figsref}[1]{\figs~\fref{#1}}
+\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
+\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
+\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
+\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
+\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
+% references to figures within bracketed text:
+\newcommand{\figb}{Fig.}
+\newcommand{\figsb}{Figs.}
+\newcommand{\figrefb}[1]{\figb~\fref{#1}}
+\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
+\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
+\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
+
+% references to tables:
+\newcommand{\tref}[1]{\textup{\ref{#1}}}
+% references to tables in normal text:
+\newcommand{\tab}{Tab.}
+\newcommand{\Tab}{Table}
+\newcommand{\tabs}{Tabs.}
+\newcommand{\Tabs}{Tables}
+\newcommand{\tabref}[1]{\tab~\tref{#1}}
+\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
+\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
+\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
+% references to tables within bracketed text:
+\newcommand{\tabb}{Tab.}
+\newcommand{\tabsb}{Tab.}
+\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
+\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
+
+
+%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\eqref}[1]{(\ref{#1})}
+\newcommand{\eqn}{Eq.}
+\newcommand{\Eqn}{Eq.}
+\newcommand{\eqns}{Eqs.}
+\newcommand{\Eqns}{Eqs.}
+\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
+\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
+\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
+\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
+
 
 %%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \usepackage{listings}
@@ -290,6 +421,36 @@ spricht von einer Wahrscheinlichkeitsdichte.
 \end{figure}
 
 
+\subsection{Korrelation}
+
+\begin{figure}[t]
+  \includegraphics[width=1\textwidth]{correlation}
+  \caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.}
+\end{figure}
+
+Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut.
+Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden.
+Der Korrelationskoeffizient 
+\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
+  (x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
+    (x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
+    \rangle)^2} \rangle} \] quantifiziert einfache lineare
+Zusammenh\"ange. Perfekt korrelierte Variablen ergeben einen
+Korrelationskoeffizienten von $+1$, antikorrelierte Daten einen
+Korrelationskoeffizienten von $-1$ und nicht korrelierte Daten einen
+Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
+
+\begin{figure}[t]
+  \includegraphics[width=1\textwidth]{nonlincorrelation}
+  \caption{\label{nonlincorrelationfig} Nichtlineare Zusammenh\"ange
+    werden durch den Korrelationskoeffizienten nicht erfasst! Sowohl
+    die quadratische Abh\"angigkeit (links) als auch eine
+    Rauschkorrelation (rechts), bei der die Streuung der $y$-Werte von
+    $x$ abh\"angen, ergeben Korrelationskeffizienten nahe Null.
+    $\xi$ sind normalverteilte Zufallszahlen.}
+\end{figure}
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Data types}
 
diff --git a/statistics/lecture/nonlincorrelation.py b/statistics/lecture/nonlincorrelation.py
new file mode 100644
index 0000000..c0ca723
--- /dev/null
+++ b/statistics/lecture/nonlincorrelation.py
@@ -0,0 +1,42 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+plt.xkcd()
+fig = plt.figure( figsize=(6,3) )
+n = 200
+x = np.random.randn( n )
+y = np.random.randn( n )
+
+z = x*x+0.2*y
+r =np.corrcoef(x,z)[0,1]
+ax = fig.add_subplot( 1, 2, 1 )
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.text( 0, 4.0, 'r=%.1f' % r, ha='center' )
+ax.text( 0, 5.5, r'$y = x^2+\xi/5$', ha='center' )
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_xlim( -3.0, 3.0)
+ax.set_ylim( -0.5, 6.0)
+ax.scatter( x, z )
+
+z = 0.5*x*y
+r =np.corrcoef(x,z)[0,1]
+ax = fig.add_subplot( 1, 2, 2 )
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.text( 0, 1.5, 'r=%.1f' % r, ha='center' )
+ax.text( 0, 2.5, r'$y = x \cdot \xi/2$', ha='center' )
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_xlim( -3.0, 3.0)
+ax.set_ylim( -3.0, 3.0)
+ax.scatter( x, z )
+
+plt.tight_layout()
+plt.savefig('nonlincorrelation.pdf')
+plt.show()