diff --git a/header.tex b/header.tex
index 4231e0c..080215b 100644
--- a/header.tex
+++ b/header.tex
@@ -216,21 +216,30 @@
 %%%%% code/matlab commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \usepackage{textcomp}
 
+% typeset code inline:
 \newcommand{\varcode}[1]{\setlength{\fboxsep}{0.5ex}\colorbox{codeback}{\texttt{#1\protect\rule[-0.1ex]{0pt}{1.6ex}}}}
 
-\newcommand{\code}[2][]{\varcode{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[mcode]{#2}}{\protect\sindex[mcode]{#1}}}
-
+% type set code and add it to the python index:
 \newcommand{\pcode}[2][]{\varcode{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[pcode]{#2}}{\protect\sindex[pcode]{#1}}}
 
+% type set code and add it to the matlab index:
 \newcommand{\mcode}[2][]{\varcode{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[mcode]{#2}}{\protect\sindex[mcode]{#1}}}
 
+% XXX typeset code and put it into matlab index: 
+% THIS SHOULD actually take both the matlab and the python code!
+\newcommand{\code}[2][]{\varcode{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[mcode]{#2}}{\protect\sindex[mcode]{#1}}}
+
+% the name of the python language:
 \newcommand{\python}{Python}
 
+% the name of the matlab language:
 \newcommand{\matlab}{\texorpdfstring{MATLAB$^{\copyright}$}{MATLAB}}
 
-\newcommand{\pythonfun}[2][]{(\tr{\python-function}{\python-Funktion} \setlength{\fboxsep}{0.5ex}\colorbox{codeback}{\texttt{#2}})\ifthenelse{\equal{#1}{}}{\protect\sindex[pcode]{#2}}{\protect\sindex[pcode]{#1}}}
+% typeset '(python-function #1)' and add the function to the python index:
+\newcommand{\pythonfun}[1]{(\tr{\python-function}{\python-Funktion} \varcode{#1})\protect\sindex[pcode]{#1}}
 
-\newcommand{\matlabfun}[2][]{(\tr{\matlab-function}{\matlab-Funktion} \setlength{\fboxsep}{0.5ex}\colorbox{codeback}{\texttt{#2}})\ifthenelse{\equal{#1}{}}{\protect\sindex[mcode]{#2}}{\protect\sindex[mcode]{#1}}}
+% typeset '(matlab-function #1)' and add the function to the matlab index:
+\newcommand{\matlabfun}[1]{(\tr{\matlab-function}{\matlab-Funktion} \varcode{#1})\protect\sindex[mcode]{#1}}
 
 
 %%%%% shortquote and widequote commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/statistics/lecture/displayunivariatedata.py b/statistics/lecture/displayunivariatedata.py
index ac62b88..546d4ff 100644
--- a/statistics/lecture/displayunivariatedata.py
+++ b/statistics/lecture/displayunivariatedata.py
@@ -1,54 +1,111 @@
 import numpy as np
 import matplotlib.pyplot as plt
 
-rng = np.random.RandomState(981)
-x = rng.randn(40, 10) + 4.0
+#rng = np.random.RandomState(981)
+#data = rng.randn(40, 1) + 4.0
+rng = np.random.RandomState(1981)
+data = rng.gamma(1.0, 1.5, 40) + 1.0
+data = data[data<7.5]
+xpos = 0.08
+ypos = 0.15
+width = 0.65
+height = 0.8
 
 plt.xkcd()
 fig = plt.figure( figsize=(6,3.4) )
-ax = fig.add_subplot(1, 1, 1)
+
+ax = fig.add_axes([xpos, ypos, width, height])
 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_xticklabels([])
+ax.set_xlim(0.0, 4.8)
 ax.set_ylabel('x')
 ax.set_ylim( 0.0, 8.0)
-ax.scatter(0.5+rng.rand(len(x[:, 5])), x[:, 5], s=50)
-ax.bar([2.0], [np.mean(x[:, 5])], 1.0, yerr=[np.std(x[:, 5])],
-       ecolor='k', capsize=0, error_kw={'elinewidth':5})
-
-## ax.annotate('Median',
-##             xy=(3.9, 0.0), xycoords='data',
-##             xytext=(3.5, -2.7), 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, -0.9), xycoords='data',
-##             xytext=(5.5, -3.4), textcoords='data', ha='right',
-##             arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0),
-##             connectionstyle="angle3,angleA=30,angleB=70") )
-## ax.annotate('3. quartile',
-##             xy=(6.1, 1.1), 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, -1.9), 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, 2.7), 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[:, 5], positions=[4.0], whis=100.0 )
-#ax.boxplot( x[:, 5], positions=[4.0], widths=[1.0], whis=100.0, manage_xticks=False )
-ax.boxplot( x[:, 5], positions=[4.0], widths=[1.0], whis=100.0 )
-ax.set_xlim(0.0, 5.0)
-ax.set_xticks([1, 3, 5], ['a', 'b', 'c'])
-plt.tight_layout()
+barwidth = 0.8
+
+scatterpos = 1.0
+barpos = 2.5
+boxpos = 4.0
+
+ax.set_xticks([scatterpos, barpos, boxpos])
+ax.set_xticklabels(['(1) data', '(2) bar\n plot', '(3) box-\nwhisker'])
+
+ax.scatter(scatterpos-0.5*barwidth+rng.rand(len(data)), data, s=50)
+
+barmean = np.mean(data)
+barstd = np.std(data)
+ew = 0.2
+ax.bar([barpos-0.5*barwidth], [barmean], barwidth, color='#FFCC00')
+eargs = {'color': 'k', 'lw': 2}
+ax.plot([barpos, barpos], [barmean-barstd, barmean+barstd], **eargs)
+ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean-barstd, barmean-barstd], **eargs)
+ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean+barstd, barmean+barstd], **eargs)
+ax.annotate('mean',
+            xy=(barpos-0.4*barwidth, 2.7), xycoords='data',
+            xytext=(barpos-1*barwidth, 5.5), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
+            connectionstyle="angle3,angleA=0,angleB=120") )
+ax.annotate('mean plus\nstd. dev.',
+            xy=(barpos+0.05*barwidth, 4.2), xycoords='data',
+            xytext=(barpos-1*barwidth, 7.0), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
+            connectionstyle="angle3,angleA=-60,angleB=80") )
+
+ax = fig.add_axes([xpos, ypos, width, height], axis_bgcolor='none')
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.spines['left'].set_visible(False)
+ax.spines['bottom'].set_visible(False)
+ax.xaxis.set_ticks_position('none')
+ax.yaxis.set_ticks_position('none')
+ax.set_xticklabels([])
+ax.set_yticklabels([])
+wh = ax.boxplot( data, positions=[boxpos], widths=[barwidth], whis=100.0, patch_artist=True)
+wh['medians'][0].set_linewidth(4)
+wh['whiskers'][0].set_linewidth(2)
+wh['whiskers'][1].set_linewidth(2)
+wh['whiskers'][0].set_linestyle('-')
+wh['whiskers'][1].set_linestyle('-')
+whiskercolor = 'k'
+wh['whiskers'][0].set_color(whiskercolor)
+wh['whiskers'][1].set_color(whiskercolor)
+wh['caps'][0].set_color(whiskercolor)
+wh['caps'][1].set_color(whiskercolor)
+wh['boxes'][0].set_facecolor('#99ff00')
+ax.set_xlim(0.0, 4.8)
+ax.set_ylim( 0.0, 8.0)
+ax.annotate('maximum',
+            xy=(boxpos, 6.5), xycoords='data',
+            xytext=(boxpos-1*barwidth, 7.6), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
+            connectionstyle="angle3,angleA=0,angleB=120") )
+ax.annotate('3. quartile',
+            xy=(boxpos-0.3*barwidth, 3.7), xycoords='data',
+            xytext=(boxpos-1.3*barwidth, 5.5), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(0.4,0.0),
+            connectionstyle="angle3,angleA=0,angleB=120") )
+ax.annotate('median',
+            xy=(boxpos+0.6*barwidth, 2.2), xycoords='data',
+            xytext=(boxpos+0.1*barwidth, 4.2), textcoords='data', ha='left',
+            arrowprops=dict(arrowstyle="->", relpos=(0.8,0.0),
+            connectionstyle="angle3,angleA=-60,angleB=20") )
+
+ax = fig.add_axes([xpos+width+0.03, ypos, 0.98-(xpos+width+0.03), height], axis_bgcolor='none')
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.xaxis.set_ticks_position('bottom')
+ax.yaxis.set_ticks_position('left')
+ax.set_yticklabels([])
+ax.set_ylim( 0.0, 8.0)
+ax.set_xticks(np.arange(0.0, 0.4, 0.1))
+ax.set_xlabel('(4) p(x)')
+bw = 0.75
+bins = np.arange(0, 8.0+bw, bw)
+h, b = np.histogram(data, bins)
+ax.barh(b[:-1], h/bw/np.sum(h), bw, color='#CC0000')
+
 plt.savefig('displayunivariatedata.pdf')
 #plt.show()
 
diff --git a/statistics/lecture/median.py b/statistics/lecture/median.py
index 9bc8bef..8204688 100644
--- a/statistics/lecture/median.py
+++ b/statistics/lecture/median.py
@@ -7,7 +7,7 @@ x = np.arange( -3.0, 3.0, 0.01 )
 g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
 
 plt.xkcd()
-fig = plt.figure( figsize=(6,3) )
+fig = plt.figure( figsize=(6, 2.8) )
 ax = fig.add_subplot(1, 2, 1)
 ax.spines['right'].set_visible(False)
 ax.spines['top'].set_visible(False)
@@ -72,6 +72,7 @@ ax.plot(x, g, 'b', lw=4)
 ax.plot([m, m], [0.0, 0.38], 'k', lw=2 )
 #ax.plot([gm, gm], [0.0, 0.42], 'k', lw=2 )
 
-plt.tight_layout()
+#plt.tight_layout()
+plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
 fig.savefig( 'median.pdf' )
 #plt.show()
diff --git a/statistics/lecture/pdfprobabilities.py b/statistics/lecture/pdfprobabilities.py
index 26c8e22..10c4482 100644
--- a/statistics/lecture/pdfprobabilities.py
+++ b/statistics/lecture/pdfprobabilities.py
@@ -30,6 +30,7 @@ ax.annotate('$P(0<x<1) = \int_0^1 p(x) \, dx$',
             connectionstyle="angle3,angleA=10,angleB=80") )
 ax.fill_between( x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], color='#cc0000' )
 ax.plot(x,g, 'b', lw=4)
-plt.tight_layout()
+#plt.tight_layout()
+plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
 fig.savefig( 'pdfprobabilities.pdf' )
 #plt.show()
diff --git a/statistics/lecture/statistics-chapter.tex b/statistics/lecture/statistics-chapter.tex
index edfdf93..e90cfe1 100644
--- a/statistics/lecture/statistics-chapter.tex
+++ b/statistics/lecture/statistics-chapter.tex
@@ -18,6 +18,7 @@
 
 \section{TODO}
 \begin{itemize}
+\item Replace exercise 1.3 (boxwhisker) by one recreating figure 1.
 \item Proper introduction to probabilities and densities first!
 \item Cumulative propability
 \item Kernel Histogramms (important for convolved PSTH)!
diff --git a/statistics/lecture/statistics.tex b/statistics/lecture/statistics.tex
index 8019fc9..ca86b2e 100644
--- a/statistics/lecture/statistics.tex
+++ b/statistics/lecture/statistics.tex
@@ -4,8 +4,8 @@
 
 Descriptive statistics characterizes data sets by means of a few measures.
 
-In addition to histograms that visualize the distribution of the data,
-the following measures are used for characterizing the univariate data:
+In addition to histograms that estimate the full distribution of the data,
+the following measures are used for characterizing univariate data:
 \begin{description}
 \item[Location, central tendency] (``Lagema{\ss}e''):
   arithmetic mean, median, mode.
@@ -20,30 +20,58 @@ For bivariate and multivariate data sets we can also analyse their
   Spearman's rank correlation coefficient.
 \end{description}
 
+The following is not a complete introduction to descriptive
+statistics, but summarizes a few concepts that are most important in
+daily data-analysis problems.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Mean, variance, and standard deviation}
 The \enterm{arithmetic mean} is a measure of location. For $n$ data values
 $x_i$ the arithmetic mean is computed by
 \[ \bar x = \langle x \rangle = \frac{1}{N}\sum_{i=1}^n x_i \; . \]
+This computation (summing up all elements of a vector and dividing by
+the length of the vector) is provided by the function \mcode{mean()}.
 The mean has the same unit as the data values.
 
 The dispersion of the data values around the mean is quantified by
 their \enterm{variance}
 \[ \sigma^2_x = \langle (x-\langle x \rangle)^2 \rangle = \frac{1}{N}\sum_{i=1}^n (x_i - \bar x)^2 \; . \]
+The variance is computed by the function \mcode{var()}.
 The unit of the variance is the unit of the data values squared.
 Therefore, variances cannot be compared to the mean or the data values
 themselves. In particular, variances cannot be used for plotting error
 bars along with the mean.
 
 The standard deviation
-\[ \sigma_x = \sqrt{\sigma^2_x} \; , \]
-however, has the same unit as the data values and can (and should) be
-used to display the dispersion of the data together withtheir mean.
+\[ \sigma_x = \sqrt{\sigma^2_x} \; , \] 
+as computed by the function \mcode{std()}, however, has the same unit
+as the data values and can (and should) be used to display the
+dispersion of the data together with their mean.
+
+The mean of a data set can be displayed by a bar-plot
+\matlabfun{bar()}. Additional errorbars \matlabfun{errobar()} can be
+used to illustrate the standard deviation of the data
+(\figref{displayunivariatedatafig} (2)).
 
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{displayunivariatedata}
-  \titlecaption{\label{displayunivariatefig} Display univariate
-    data.}{Bla.}
+  \titlecaption{\label{displayunivariatedatafig} Displaying statistics
+    of univariate data.}{(1) In particular for small data sets it is
+    most informative to plot the data themselves. The value of each
+    data point is plotted on the y-axis.  To make the data points
+    overlap less, they are jittered along the x-axis by means of
+    uniformly distributed random numbers \matlabfun{rand()}. (2) With
+    a bar plot \matlabfun{bar()} one usually shows the mean of the
+    data. The additional errorbar illustrates the deviation of the
+    data from the mean by $\pm$ one standard deviation. (3) A
+    box-whisker plot \matlabfun{boxplot()} shows more details of the
+    distribution of the data values. The box extends from the 1. to
+    the 3. quartile, a horizontal ine within the box marks the median
+    value, and the whiskers extend to the minum and the maximum data
+    values. (4) The probability density $p(x)$ estimated from a
+    normalized histogram shows the entire distribution of the
+    data. Estimating the probability distribution is only meaningful
+    for sufficiently large data sets.}
 \end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -54,7 +82,7 @@ used to display the dispersion of the data together withtheir mean.
   \titlecaption{\label{medianfig} Median, mean and mode of a
     probability distribution.}{Left: Median, mean and mode are
     identical for the symmetric and unimodal normal distribution.
-    Right: for asymmetric distributions these threa measures differ. A
+    Right: for asymmetric distributions these three measures differ. A
     heavy tail of a distribution pulls out the mean most strongly. In
     contrast, the median is more robust against heavy tails, but not
     necessarily identical with the mode.}
@@ -66,7 +94,6 @@ The \enterm{median} separates a list of data values into two halves
 such that one half of the data is not greater and the other half is
 not smaller than the median (\figref{medianfig}).
 
-\newpage
 \begin{exercise}{mymedian.m}{}
   Write a function \code{mymedian()} that computes the median of a vector.
 \end{exercise}
@@ -77,7 +104,11 @@ not smaller than the median (\figref{medianfig}).
   Write a script that tests whether your median function really
   returns a median above which are the same number of data than
   below. In particular the script should test data vectors of
-  different length.
+  different length. You should not use the \mcode{median()} function
+  for testing your function.
+
+  Writing tests for your own functions is a very important strategy for
+  writing reliable code!
 \end{exercise}
 
 \begin{figure}[t]
@@ -103,19 +134,19 @@ data are smaller than the 3$^{\rm rd}$ quartile.
 %   Write a function that computes the first, second, and third quartile of a vector.
 % \end{exercise}
 
-\begin{figure}[t]
-  \includegraphics[width=1\textwidth]{boxwhisker}
-  \titlecaption{\label{boxwhiskerfig} Box-Whisker Plot.}{Box-whisker
-    plots are well suited for comparing unimodal distributions.  Each
-    box-whisker characterizes 40 random numbers that have been drawn
-    from a normal distribution.}
-\end{figure}
+% \begin{figure}[t]
+%   \includegraphics[width=1\textwidth]{boxwhisker}
+%   \titlecaption{\label{boxwhiskerfig} Box-Whisker Plot.}{Box-whisker
+%     plots are well suited for comparing unimodal distributions.  Each
+%     box-whisker characterizes 40 random numbers that have been drawn
+%     from a normal distribution.}
+% \end{figure}
 
 \enterm{Box-whisker plots} are commonly used to visualize and compare
-the distribution of unimodal data. Aa box is drawn around the median
+the distribution of unimodal data. A box is drawn around the median
 that extends from the 1$^{\rm st}$ to the 3$^{\rm rd}$ quartile. The
 whiskers mark the minimum and maximum value of the data set
-(\figref{boxwhiskerfig}).
+(\figref{displayunivariatedatafig} (3)).
 
 \begin{exercise}{boxwhisker.m}{}
   Generate eine $40 \times 10$ matrix of random numbers and
@@ -123,13 +154,17 @@ whiskers mark the minimum and maximum value of the data set
   (\code{boxplot()} function). How to interpret the plot?
 \end{exercise}
 
-\section{Histograms}
+\section{Distributions}
+The distribution of values in a data set is estimated by histograms
+(\figref{displayunivariatedatafig} (4)).
+
+\subsection{Histograms}
 
 \enterm[Histogram]{Histograms} count the frequency $n_i$ of
-$N=\sum_{i=1}^M n_i$ measurements in $M$ bins $i$.  The bins tile the
-data range usually into intervals of the same size. Histograms are
-often used to estimate the \enterm{probability distribution} of the
-data values.
+$N=\sum_{i=1}^M n_i$ measurements in each of $M$ bins $i$
+(\figref{diehistogramsfig} left).  The bins tile the data range
+usually into intervals of the same size. The width of the bins is
+called the bin width.
 
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{diehistograms}
@@ -141,14 +176,33 @@ data values.
     with the expected theoretical distribution of $P=1/6$.}
 \end{figure}
 
-For integer data values (e.g. die number of the faces of a die or the
-number of action potential occurring within a fixed time window) a bin
-can be defined for each data value.  The histogram is usually
-normalized by the total number of measurements to make it
-independent of size of the data set (\figref{diehistogramsfig}). Then
-the height of each histogram bar equals the probability $P(x_i)$ of
-the data value $x_i$ in the $i$-th bin:
-\[ P(x_i) = P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \]
+Histograms are often used to estimate the \enterm{probability
+  distribution} of the data values.
+
+\subsection{Probabilities}
+In the frequentist interpretation of probability, the probability of
+an event (e.g. getting a six when rolling a die) is the relative
+occurrence of this event in the limit of a large number of trials.
+
+For a finite number of trials $N$ where the event $i$ occurred $n_i$
+times, the probability $P_i$ of this event is estimated by
+\[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \] 
+From this definition it follows that a probability is a unitless
+quantity that takes on values between zero and one. Most importantly,
+the sum of the probabilities of all possible events is one:
+\[ \sum_{i=1}^M P_i = \sum_{i=1}^M \frac{n_i}{N} = \frac{1}{N} \sum_{i=1}^M n_i = \frac{N}{N} = 1\; , \]
+i.e. the probability of getting any event is one.
+
+
+\subsection{Probability distributions of categorial data}
+
+For categorial data values (e.g. the faces of a die (as integer
+numbers or as colors)) a bin can be defined for each category $i$.
+The histogram is normalized by the total number of measurements to
+make it independent of the size of the data set
+(\figref{diehistogramsfig}). After this normalization the height of
+each histogram bar is an estimate of the probability $P_i$ of the
+category $i$, i.e. of getting a data value in the $i$-th bin.
 
 \begin{exercise}{rollthedie.m}{}
   Write a function that simulates rolling a die $n$ times.
@@ -162,38 +216,47 @@ the data value $x_i$ in the $i$-th bin:
 \end{exercise}
 
 
-\section{Probability density functions}
-
-Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun
-(z.B. Gewicht von Tigern, L\"ange von Interspikeintervallen).  Es
-macht keinen Sinn dem Auftreten jeder einzelnen reelen Zahl eine
-Wahrscheinlichkeit zuzuordnen, denn die Wahrscheinlichkeit genau den
-Wert einer bestimmten reelen Zahl, z.B. 1.23456789, zu messen ist
-gleich Null, da es unabz\"ahlbar viele reelle Zahlen gibt.
-
-Sinnvoller ist es dagegen, nach der Wahrscheinlichkeit zu fragen, eine
-Zahl aus einem bestimmten Bereich zu erhalten, z.B. die
-Wahrscheinlichkeit $P(1.2<x<1.3)$, dass die Zahl $x$ einen Wert
-zwischen 1.2 und 1.3 hat.
-
-Im Grenzwert zu sehr kleinen Bereichen $\Delta x$ ist die Wahrscheinlichkeit
-eines Wertes $x$ zwischen $x_0$ und $x_0+\Delta x$
-\[ P(x_0<x<x_0+\Delta x) \approx p(x) \cdot \Delta x \; . \] 
-Die Gr\"o{\ss}e $p(x)$ ist eine sogenannte
-\determ{Wahrscheinlichkeitsdichte}. Sie ist keine einheitenlose
-Wahrscheinlichkeit mit Werten zwischen Null und Eins, sondern kann
-jeden positiven Wert annehmen und hat als Einheit den Kehrwert der
-Einheit von $x$.
+\subsection{Probability densities functions}
+
+In cases where we deal with data sets of measurements of a real
+quantity (e.g. the length of snakes, the weight of elephants, the time
+between succeeding spikes) there is no natural bin width for computing
+a histogram. In addition, the probability of measuring a data value that
+equals exactly a specific real number like, e.g., 0.123456789 is zero, because
+there are uncountable many real numbers.
+
+We can only ask for the probability to get a measurement value in some
+range.  For example, we can ask for the probability $P(1.2<x<1.3)$ to
+get a measurement between 0 and 1 (\figref{pdfprobabilitiesfig}). More
+generally, we want to know the probability $P(x_0<x<x_1)$ to obtain a
+measurement between $x_0$ and $x_1$. If we define the width of the
+range defined by $x_0$ and $x_1$ is $\Delta x = x_1 - x_0$ then the
+probability can also be expressed as $P(x_0<x<x_0 + \Delta x)$.
+
+In the limit to very small ranges $\Delta x$ the probability of
+getting a measurement between $x_0$ and $x_0+\Delta x$ scales down to
+zero with $\Delta x$:
+\[ P(x_0<x<x_0+\Delta x) \approx p(x_0) \cdot \Delta x \; . \] 
+In here the quantity $p(x_00)$ is a so called \enterm{probability
+  density}. This is not a unitless probability with values between 0
+and 1, but a number that takes on any positive real number and has as
+a unit the inverse of the unit of the data values --- hence the name
+``density''.
 
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{pdfprobabilities}
-  \titlecaption{\label{pdfprobabilitiesfig} Wahrscheinlichkeiten bei
-  einer Wahrscheinlichkeitsdichtefunktion.}{}
+  \titlecaption{\label{pdfprobabilitiesfig} Probability of a
+    probability density.}{The probability of a data value $x$ between,
+    e.g., zero and one is the integral (red area) over the probability
+    density (blue).}
 \end{figure}
 
-F\"ur beliebige Bereiche ist die Wahrscheinlichkeit f\"ur den Wert $x$ zwischen
-$x_1$ und $x_2$ gegeben durch
+The probability to get a value $x$ between $x_1$ and $x_2$ is 
+given by the integral over the probability density:
 \[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \; . \]
+Because the probability to get any value $x$ is one, the integral over
+the probability density 
+
 Da die Wahrscheinlichkeit irgendeines Wertes $x$ Eins ergeben muss gilt die Normierung
 \begin{equation}
   \label{pdfnorm}
@@ -215,35 +278,35 @@ Standardabweichung $\sigma$.
 
 \begin{exercise}{gaussianpdf.m}{gaussianpdf.out}
   \begin{enumerate}
-  \item Plotte die Wahrscheinlichkeitsdichte der Normalverteilung $p_g(x)$.
-  \item Berechne f\"ur die Normalverteilung mit Mittelwert Null und
-    Standardabweichung Eins die Wahrscheinlichkeit, eine Zahl zwischen
-    0 und 1 zu erhalten.
-  \item Ziehe 1000 normalverteilte Zufallszahlen und bestimme von
-    diesen Zufallzahlen die Wahrscheinlichkeit der Zahlen zwischen
-    Null und Eins.
-  \item Berechne aus der Normalverteilung $\int_{-\infty}^{+\infty} p(x) \, dx$.
+  \item Plot the probability density of the normal distribution $p_g(x)$.
+  \item Compute the probability of getting a data value between zero and one 
+    for the normal distribution with zero mean and standard deviation of one.
+  \item Draw 1000 normally distributed random numbers and use these
+    numbers to calculate the probability of getting a number between
+    zero and one.
+  \item Compute from the normal distribution $\int_{-\infty}^{+\infty} p(x) \, dx$.
   \end{enumerate}
 \end{exercise}
 
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{pdfhistogram}
-  \titlecaption{\label{pdfhistogramfig} Histogramme mit verschiedenen
-    Klassenbreiten von normalverteilten Messwerten.}{Links: Die H\"ohe
-    des absoluten Histogramms h\"angt von der Klassenbreite
-    ab. Rechts: Bei auf das Integral normierten Histogrammen werden
-    auch unterschiedliche Klassenbreiten untereinander vergleichbar
-    und auch mit der theoretischen Wahrschinlichkeitsdichtefunktion
-    (blau).}
+  \titlecaption{\label{pdfhistogramfig} Histograms with different bin
+    widths of normally distributed data.}{Left: The height of the
+    histogram bars strongly depends on the width of the bins. Right:
+    If the histogram is normalized such that its integral is one we
+    get an estimate of the probability density of the data values.
+    The normalized histograms are comparable with each other and can
+    also be compared to theoretical probability densities, like the
+    normal distributions (blue).}
 \end{figure}
 
+\pagebreak[4]
 \begin{exercise}{gaussianbins.m}{}
   Draw 100 random data from a Gaussian distribution and plot
   histograms with different bin sizes of the data. What do you
   observe?
 \end{exercise}
 
-\pagebreak[2]
 Damit Histogramme von reellen Messwerten trotz unterschiedlicher
 Anzahl von Messungen und unterschiedlicher Klassenbreiten
 untereinander vergleichbar werden und mit bekannten
@@ -262,7 +325,6 @@ und das normierte Histogramm hat die H\"ohe
 Es muss also nicht nur durch die Summe, sondern auch durch die Breite
 $\Delta x$ der Klassen geteilt werden (\figref{pdfhistogramfig}).
 
-\pagebreak[4]
 \begin{exercise}{gaussianbinsnorm.m}{}
   Normiere das Histogramm der vorherigen \"Ubung zu einer Wahrscheinlichkeitsdichte.
 \end{exercise}