diff --git a/statistics/exercises/bootstrapmean.m b/statistics/exercises/bootstrapmean.m
new file mode 100644
index 0000000..356c531
--- /dev/null
+++ b/statistics/exercises/bootstrapmean.m
@@ -0,0 +1,17 @@
+function [bootsem, mu] = bootstrapmean( x, resample )
+% computes standard error by bootstrapping the data
+% x: vector with data
+% resample: number of resamplings
+% returns:
+% bootsem: the standard error of the mean
+% mu: the bootstrapped means as a vector
+ mu = zeros( resample, 1 );
+ nsamples = length(x);
+ for i = 1:resample
+ % resample:
+ xr = x(randi(nsamples, nsamples, 1));
+ % compute statistics on sample:
+ mu(i) = mean(xr);
+ end
+ bootsem = std( mu );
+end
diff --git a/statistics/exercises/bootstraptymus-datahist.pdf b/statistics/exercises/bootstraptymus-datahist.pdf
new file mode 100644
index 0000000..4cc71ac
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+++ b/statistics/exercises/bootstraptymus-datahist.pdf
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diff --git a/statistics/exercises/bootstraptymus-meanhist.pdf b/statistics/exercises/bootstraptymus-meanhist.pdf
new file mode 100644
index 0000000..f503f5f
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diff --git a/statistics/exercises/bootstraptymus-samples.pdf b/statistics/exercises/bootstraptymus-samples.pdf
new file mode 100644
index 0000000..e4d69bc
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diff --git a/statistics/exercises/bootstraptymus.m b/statistics/exercises/bootstraptymus.m
index e92a5b7..0b4aaa7 100644
--- a/statistics/exercises/bootstraptymus.m
+++ b/statistics/exercises/bootstraptymus.m
@@ -1,24 +1,47 @@
-resample = 500
-
+%% (b) load the data:
load( 'thymusglandweights.dat' );
-x = thymusglandweights;
-nsamples = length( x );
+nsamples = 80;
+x = thymusglandweights(1:nsamples);
+
+%% (c) mean, sem and hist:
sem = std(x)/sqrt(nsamples);
+fprintf( 'Mean of the data set = %.2fmg\n', mean(x) );
+fprintf( 'SEM of the data set = %.2fmg\n', sem );
+hist(x,20)
+xlabel('x')
+ylabel('count')
+savefigpdf( gcf, 'bootstraptymus-datahist.pdf', 6, 5 );
+pause( 2.0 )
-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 );
+%% (d) bootstrap the mean:
+resample = 500;
+[bootsem, mu] = bootstrapmean( x, resample );
+hist( mu, 20 );
+xlabel('mean(x)')
+ylabel('count')
+savefigpdf( gcf, 'bootstraptymus-meanhist.pdf', 6, 5 );
+fprintf( ' bootstrap standard error: %.3f\n', bootsem );
+fprintf( 'theoretical standard error: %.3f\n', sem );
+%% (e) confidence interval:
+q = quantile(mu, [0.025, 0.975]);
+fprintf( '95%% confidence interval of the mean from %.2fmg to %.2fmg\n', q(1), q(2) );
+pause( 2.0 )
+
+%% (f): dependence on sample size:
+nsamplesrange = 10:10:1000;
+bootsems = zeros( length(nsamplesrange),1);
+for n=1:length(nsamplesrange)
+ nsamples = nsamplesrange(n);
+ % [bootsems(n), mu] = bootstrapmean(x, resample);
+ bootsems(n) = bootstrapmean(thymusglandweights(1:nsamples), resample);
+end
+plot(nsamplesrange, bootsems, 'b', 'linewidth', 2);
hold on
-hist( x, 20 );
-hist( mu, 20 );
+plot(nsamplesrange, std(x)./sqrt(nsamplesrange), 'r', 'linewidth', 1)
hold off
-
-disp(['bootstrap standard error: ', num2str(bootsem)]);
-disp(['standard error: ', num2str(sem)]);
+xlabel('sample size')
+ylabel('SEM')
+legend('bootsrap', 'theory')
+savefigpdf( gcf, 'bootstraptymus-samples.pdf', 6, 5 );
diff --git a/statistics/exercises/centrallimit-hist01.pdf b/statistics/exercises/centrallimit-hist01.pdf
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diff --git a/statistics/exercises/centrallimit-hist06.pdf b/statistics/exercises/centrallimit-hist06.pdf
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diff --git a/statistics/exercises/centrallimit-hist07.pdf b/statistics/exercises/centrallimit-hist07.pdf
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diff --git a/statistics/exercises/centrallimit-hist08.pdf b/statistics/exercises/centrallimit-hist08.pdf
new file mode 100644
index 0000000..3a7db93
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diff --git a/statistics/exercises/centrallimit-hist09.pdf b/statistics/exercises/centrallimit-hist09.pdf
new file mode 100644
index 0000000..e0b7ec7
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diff --git a/statistics/exercises/centrallimit-hist10.pdf b/statistics/exercises/centrallimit-hist10.pdf
new file mode 100644
index 0000000..e10fee8
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diff --git a/statistics/exercises/centrallimit-samples.pdf b/statistics/exercises/centrallimit-samples.pdf
new file mode 100644
index 0000000..cef3be6
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diff --git a/statistics/exercises/centrallimit.m b/statistics/exercises/centrallimit.m
index 9f492bd..0794335 100644
--- a/statistics/exercises/centrallimit.m
+++ b/statistics/exercises/centrallimit.m
@@ -2,7 +2,7 @@
n = 10000;
m = 10; % number of loops
-%% (b) a single random number:
+%% (b) a single data set of random numbers:
x = rand( n, 1 );
%% (c) plot probability density:
@@ -41,7 +41,7 @@ for i=1:m
%xx = min(x):0.01:max(x);
xx = -1:0.01:i+1; % x-axis values for plot of pdf
p = exp(-0.5*(xx-mu).^2/sd^2)/sqrt(2*pi*sd^2); % pdf
- plot(xx, p, 'r', 'linewidth', 6 )
+ plot(xx, p, 'r', 'linewidth', 3 )
ns = sprintf( 'N=%d', i );
text( 0.1, 0.9, ns, 'units', 'normalized' )
hold on
@@ -50,8 +50,10 @@ for i=1:m
h = h/sum(h)/(b(2)-b(1)); % normalization
bar(b, h)
hold off
+ xlim([-0.5, i+0.5])
xlabel( 'x' )
ylabel( 'summed pdf' )
+ savefigpdf( gcf, sprintf('centrallimit-hist%02d.pdf', i), 6, 5 );
if i < 6
pause( 3.0 )
end
@@ -68,5 +70,6 @@ plot( xx, sqrt(xx)*sdu, 'k' )
legend( 'mean', 'std', 'theory' )
xlabel('N')
hold off
+savefigpdf( gcf, 'centrallimit-samples.pdf', 6, 5 );
diff --git a/statistics/exercises/correlationsignificance.m b/statistics/exercises/correlationsignificance.m
index 7a0f5f0..97c09e7 100644
--- a/statistics/exercises/correlationsignificance.m
+++ b/statistics/exercises/correlationsignificance.m
@@ -6,22 +6,27 @@ y = randn(n, 1) + a*x;
%% (b) scatter plot:
subplot(1, 2, 1);
+plot(x, a*x, 'r', 'linewidth', 4 );
+hold on
%scatter(x, y ); % either scatter ...
plot(x, y, 'o' ); % ... or plot - same plot.
+xlabel('x')
+ylabel('y')
+hold off
%% (d) correlation coefficient:
+%c = corrcoef(x, y); % returns correlation matrix
+%rd = c(1, 2);
rd = corr(x, y);
-%rd = r(0, 1);
fprintf('correlation coefficient = %.2f\n', rd );
-%% (f) permutation:
+%% (e) permutation:
nperm = 1000;
rs = zeros(nperm,1);
for i=1:nperm
xr=x(randperm(length(x))); % shuffle x
yr=y(randperm(length(y))); % shuffle y
rs(i) = corr(xr, yr);
- %rs(i) = r(0,1);
end
%% (g) pdf of the correlation coefficients:
@@ -44,6 +49,7 @@ bar(b, h, 'facecolor', 'b');
bar(b(b>=rq), h(b>=rq), 'facecolor', 'r');
plot( [rd rd], [0 4], 'r', 'linewidth', 2 );
xlabel('correlation coefficient');
-ylabel('probability density');
+ylabel('probability density of H0');
hold off;
+savefigpdf( gcf, 'correlationsignificance.pdf', 12, 7 );
diff --git a/statistics/exercises/die1.m b/statistics/exercises/die1.m
index d407ff1..32ddad4 100644
--- a/statistics/exercises/die1.m
+++ b/statistics/exercises/die1.m
@@ -18,10 +18,11 @@ for i =1:6
P(i) = sum(x == i)/length(x);
end
subplot( 1, 2, 1 )
-plot( [0 7], [1/6 1/6], 'r', 'linewidth', 6 )
+plot( [0 7], [1/6 1/6], 'r', 'linewidth', 3 )
hold on
bar( P );
hold off
+set(gca, 'XTick', 1:6 );
xlim( [ 0 7 ] );
xlabel('Eyes');
ylabel('Probability');
@@ -35,5 +36,4 @@ diehist( x );
x = randi( 8, 1, n ); % random numbers from 1 to 8
x(x>6) = 6; % set numbers 7 and 8 to 6
diehist( x );
-
-
+savefigpdf(gcf, 'die1.pdf', 12, 5)
\ No newline at end of file
diff --git a/statistics/exercises/die1.pdf b/statistics/exercises/die1.pdf
new file mode 100644
index 0000000..bc4ed16
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diff --git a/statistics/exercises/die2.m b/statistics/exercises/die2.m
index 407592b..bd2e5f1 100644
--- a/statistics/exercises/die2.m
+++ b/statistics/exercises/die2.m
@@ -17,8 +17,10 @@ s = std(P, 1);
bar(m, 'facecolor', [0.8 0 0]); % darker red
hold on;
errorbar(m, s, '.k', 'linewidth', 2 ); % k is black
+set(gca, 'XTick', 1:6 );
xlim( [ 0, 7 ] );
+ylim( [ 0, 0.25])
xlabel('Eyes');
ylabel('Probability');
hold off;
-
+savefigpdf(gcf, 'die2.pdf', 6, 5)
diff --git a/statistics/exercises/die2.pdf b/statistics/exercises/die2.pdf
new file mode 100644
index 0000000..1bd4ec6
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diff --git a/statistics/exercises/diehist.m b/statistics/exercises/diehist.m
index 4a8c24a..1581884 100644
--- a/statistics/exercises/diehist.m
+++ b/statistics/exercises/diehist.m
@@ -3,10 +3,11 @@ function diehist( x )
% die.
[h,b] = hist( x, 1:6 );
h = h/sum(h); % normalization
- plot( [0 7], [1/6 1/6], 'r', 'linewidth', 6 )
+ plot( [0 7], [1/6 1/6], 'r', 'linewidth', 3 )
hold on
bar( b, h );
hold off
+ set(gca, 'XTick', 1:6 );
xlim( [ 0, 7 ] );
xlabel('Eyes');
ylabel('Probability');
diff --git a/statistics/exercises/normprobs.m b/statistics/exercises/normprobs.m
index c7e4f37..d858b27 100644
--- a/statistics/exercises/normprobs.m
+++ b/statistics/exercises/normprobs.m
@@ -27,17 +27,20 @@ fprintf( 'Integral over the Gaussian pdf from -3 to 3 is %.4f\n\n', P );
%% (e) probability of small ranges
nr = 50;
+xmax = 3.0
xs = zeros(nr, 1); % size of integration interval
Ps = zeros(nr, 1); % storage
for i = 1:nr
% upper limit goes from 4.0 down to 0.0:
- xupper = 3.0*(nr-i)/nr;
+ xupper = xmax*(nr-i)/nr;
xs(i) = xupper;
% integral from 0 to xupper:
Ps(i) = sum(pg((xx>=0.0)&(xx<=xupper)))*dx;
end
plot( xs, Ps, 'linewidth', 3 )
+xlim([0 xmax])
ylim([0 0.55])
xlabel('Integration interval')
ylabel('Probability')
fprintf('The probability P(0.1234) = %.4f\n\n', sum(x == 0.1234)/length(x) );
+savefigpdf(gcf, 'normprobs.pdf', 12, 8);
diff --git a/statistics/exercises/normprobs.pdf b/statistics/exercises/normprobs.pdf
new file mode 100644
index 0000000..f0e5456
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diff --git a/statistics/exercises/randomwalk-hists.pdf b/statistics/exercises/randomwalk-hists.pdf
new file mode 100644
index 0000000..4da23c9
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diff --git a/statistics/exercises/randomwalk-stdev.pdf b/statistics/exercises/randomwalk-stdev.pdf
new file mode 100644
index 0000000..33cd212
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diff --git a/statistics/exercises/randomwalk-traces.pdf b/statistics/exercises/randomwalk-traces.pdf
new file mode 100644
index 0000000..4857900
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diff --git a/statistics/exercises/randomwalkstatistics.m b/statistics/exercises/randomwalkstatistics.m
index 8653aa6..e0c2946 100644
--- a/statistics/exercises/randomwalkstatistics.m
+++ b/statistics/exercises/randomwalkstatistics.m
@@ -11,9 +11,10 @@ for i=1:length(nwalks)
end
text( 0.05, 0.8, sprintf( 'N=%d', nwalks(i)), 'units', 'normalized' )
xlabel( 'Number of steps' );
- ylabel( 'Position of walker' )
+ ylabel( 'Position' )
hold off;
end
+savefigpdf( gcf, 'randomwalk-traces.pdf', 12, 16 );
pause( 5.0 )
nsteps = 100;
@@ -34,7 +35,10 @@ xx = 0:0.01:nsteps;
plot( xx, sqrt(xx), 'k' )
plot( xx, zeros(length(xx),1), 'k' )
legend( 'mean', 'std', 'theory' )
+xlabel('Steps')
+ylabel('Position')
hold off
+savefigpdf( gcf, 'randomwalk-stdev.pdf', 6, 5 );
pause( 3.0 );
%% (d) histograms:
@@ -47,5 +51,8 @@ for i = 1:length(tinx)
hold on;
end
hold off;
-xlabel('Position of walker');
+xlabel('Position');
ylabel('Probability density');
+xlim([-30 30])
+ylim([0 0.3])
+savefigpdf( gcf, 'randomwalk-hists.pdf', 6, 5 );
diff --git a/statistics/exercises/savefigpdf.m b/statistics/exercises/savefigpdf.m
new file mode 100644
index 0000000..fbe2dc2
--- /dev/null
+++ b/statistics/exercises/savefigpdf.m
@@ -0,0 +1,28 @@
+function savefigpdf( fig, name, width, height )
+% Saves figure fig in pdf file name.pdf with appropriately set page size
+% and fonts
+
+% default width:
+if nargin < 3
+ width = 11.7;
+end
+% default height:
+if nargin < 4
+ height = 9.0;
+end
+
+% paper:
+set( fig, 'PaperUnits', 'centimeters' );
+set( fig, 'PaperSize', [width height] );
+set( fig, 'PaperPosition', [0.0 0.0 width height] );
+set( fig, 'Color', 'white')
+
+% font:
+set( findall( fig, 'type', 'axes' ), 'FontSize', 12 )
+set( findall( fig, 'type', 'text' ), 'FontSize', 12 )
+
+% save:
+saveas( fig, name, 'pdf' )
+
+end
+
diff --git a/statistics/exercises/statistics01.tex b/statistics/exercises/statistics01.tex
index d78a0b3..767142c 100644
--- a/statistics/exercises/statistics01.tex
+++ b/statistics/exercises/statistics01.tex
@@ -115,6 +115,7 @@ Der Computer kann auch als W\"urfel verwendet werden!
\lstinputlisting{rollthedie.m}
\lstinputlisting{diehist.m}
\lstinputlisting{die1.m}
+ \includegraphics[width=1\textwidth]{die1}
\end{solution}
@@ -132,6 +133,7 @@ Wir werten nun das Verhalten mehrerer W\"urfel aus.
\end{parts}
\begin{solution}
\lstinputlisting{die2.m}
+ \includegraphics[width=0.5\textwidth]{die2}
\end{solution}
@@ -173,6 +175,7 @@ Mittelwert enthalten ist.
\end{parts}
\begin{solution}
\lstinputlisting{normprobs.m}
+ \includegraphics[width=1\textwidth]{normprobs}
\end{solution}
diff --git a/statistics/exercises/statistics02.tex b/statistics/exercises/statistics02.tex
index 36216be..f2acd45 100644
--- a/statistics/exercises/statistics02.tex
+++ b/statistics/exercises/statistics02.tex
@@ -121,6 +121,11 @@ Den Zentralen Grenzwertsatz wollen wir uns im Folgenden veranschaulichen.
\end{parts}
\begin{solution}
\lstinputlisting{centrallimit.m}
+ \includegraphics[width=0.5\textwidth]{centrallimit-hist01}
+ \includegraphics[width=0.5\textwidth]{centrallimit-hist02}
+ \includegraphics[width=0.5\textwidth]{centrallimit-hist03}
+ \includegraphics[width=0.5\textwidth]{centrallimit-hist05}
+ \includegraphics[width=0.5\textwidth]{centrallimit-samples}
\end{solution}
@@ -147,6 +152,9 @@ Im folgenden wollen wir einige Eigenschaften des Random Walks bestimmen.
\begin{solution}
\lstinputlisting{randomwalk.m}
\lstinputlisting{randomwalkstatistics.m}
+ \includegraphics[width=0.8\textwidth]{randomwalk-traces}\\
+ \includegraphics[width=0.5\textwidth]{randomwalk-stdev}
+ \includegraphics[width=0.5\textwidth]{randomwalk-hists}
\end{solution}
diff --git a/statistics/exercises/statistics03.tex b/statistics/exercises/statistics03.tex
index 6ae5189..c80a013 100644
--- a/statistics/exercises/statistics03.tex
+++ b/statistics/exercises/statistics03.tex
@@ -103,19 +103,33 @@ jan.benda@uni-tuebingen.de}
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}
+\begin{solution}
+ \lstinputlisting{bootstrapmean.m}
+ \lstinputlisting{bootstraptymus.m}
+ \includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
+ \includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
+ \includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
+\end{solution}
\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 Ziehe daraus 1000 Stichproben vom Umfang $m=3$, 5, 10, oder 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 $t=\bar x/(\sigma_x/\sqrt{m}$
+\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}
+\begin{solution}
+ \lstinputlisting{tdistribution.m}
+ \includegraphics[width=1\textwidth]{tdistribution-n03}\\
+ \includegraphics[width=1\textwidth]{tdistribution-n05}\\
+ \includegraphics[width=1\textwidth]{tdistribution-n10}\\
+ \includegraphics[width=1\textwidth]{tdistribution-n50}
+\end{solution}
\question \qt{Korrelationen}
@@ -135,8 +149,13 @@ 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.
+\part Variiere die Stichprobengr\"o{\ss}e \code{n} und \"uberpr\"ufe
+auf gleiche Weise die Signifikanz.
\end{parts}
+\begin{solution}
+ \lstinputlisting{correlationsignificance.m}
+ \includegraphics[width=1\textwidth]{correlationsignificance}
+\end{solution}
\end{questions}
diff --git a/statistics/exercises/tdistribution-n03.pdf b/statistics/exercises/tdistribution-n03.pdf
new file mode 100644
index 0000000..8d780be
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diff --git a/statistics/exercises/tdistribution-n05.pdf b/statistics/exercises/tdistribution-n05.pdf
new file mode 100644
index 0000000..b69abeb
Binary files /dev/null and b/statistics/exercises/tdistribution-n05.pdf differ
diff --git a/statistics/exercises/tdistribution-n10.pdf b/statistics/exercises/tdistribution-n10.pdf
new file mode 100644
index 0000000..bd77f2e
Binary files /dev/null and b/statistics/exercises/tdistribution-n10.pdf differ
diff --git a/statistics/exercises/tdistribution-n50.pdf b/statistics/exercises/tdistribution-n50.pdf
new file mode 100644
index 0000000..f3c4fe5
Binary files /dev/null and b/statistics/exercises/tdistribution-n50.pdf differ
diff --git a/statistics/exercises/tdistribution.m b/statistics/exercises/tdistribution.m
index 9d8920b..223cbe5 100644
--- a/statistics/exercises/tdistribution.m
+++ b/statistics/exercises/tdistribution.m
@@ -1,32 +1,58 @@
-n = 100000
+%% (a) generate random numbers:
+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);
+for nsamples=[3 5 10 50]
+ nsamples
+ %% compute mean, standard deviation and t:
+ 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
+
+ % Gaussian pdfs:
+ msdev = std(means);
+ tsdev = 1.0;
+ dxg=0.01;
+ xmax = 10.0;
+ xmin = -xmax;
+ xg = [xmin:dxg:xmax];
+ pm = exp(-0.5*(xg/msdev).^2)/sqrt(2.0*pi)/msdev;
+ pt = exp(-0.5*(xg/tsdev).^2)/sqrt(2.0*pi)/tsdev;
+
+ %% plots
+ subplot(1, 2, 1)
+ bins = xmin:0.2:xmax;
+ [h,b] = hist(means, bins);
+ h = h/sum(h)/(b(2)-b(1));
+ bar(b, h, 'facecolor', 'b', 'edgecolor', 'b')
+ hold on
+ plot(xg, pm, 'r', 'linewidth', 2)
+ title( sprintf('sample size = %d', nsamples) );
+ xlim( [-3, 3] );
+ xlabel('Mean');
+ ylabel('pdf');
+ hold off;
+
+ subplot(1, 2, 2)
+ bins = xmin:0.5:xmax;
+ [h,b] = hist(students, bins);
+ h = h/sum(h)/(b(2)-b(1));
+ bar(b, h, 'facecolor', 'b', 'edgecolor', 'b')
+ hold on
+ plot(xg, pt, 'r', 'linewidth', 2)
+ title( sprintf('sample size = %d', nsamples) );
+ xlim( [-8, 8] );
+ xlabel('Student-t');
+ ylabel('pdf');
+ hold off;
+
+ savefigpdf( gcf, sprintf('tdistribution-n%02d.pdf', nsamples), 14, 5 );
+ pause( 3.0 )
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;
-