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