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Jan Benda
2018-11-12 14:18:18 +01:00
parent 67ef51356e
commit 665231c00c
7 changed files with 147 additions and 49 deletions
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118
statistics/exercises/exercises01.tex
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@@ -15,7 +15,7 @@
\else
\newcommand{\stitle}{}
\fi
\header{{\bfseries\large Exercise 6\stitle}}{{\bfseries\large Statistics}}{{\bfseries\large November 14th, 2017}}
\header{{\bfseries\large Exercise 7\stitle}}{{\bfseries\large Statistics}}{{\bfseries\large November 13th, 2018}}
\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
jan.benda@uni-tuebingen.de}
\runningfooter{}{\thepage}{}
@@ -104,7 +104,7 @@ jan.benda@uni-tuebingen.de}
addition, the internet offers a lot of material and suggestions for
any question you have regarding your code !
\item Please upload your solution to the exercises to ILIAS as a zip-archive with the name
``probabilities\_\{last name\}\_\{first name\}.zip''.
``statistics\_\{last name\}\_\{first name\}.zip''.
\end{itemize}
\fi
@@ -116,7 +116,7 @@ jan.benda@uni-tuebingen.de}
\question \textbf{Read chapter 4 of the script on ``programming style''!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Probabilities of a die I}
\question \qt{Probabilities of a die}
The computer can roll dice with more than 6 faces!
\begin{parts}
\part Simulate 10000 times rolling a die with eight faces by
@@ -152,22 +152,6 @@ The computer can roll dice with more than 6 faces!
\continue
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Probabilities of a die II}
Now we analyze several dice at once.
\begin{parts}
\part Simulate 20 dice, each of which is rolled 100 times
(each die is simulated with the same random number generator).
\part Compute for this data set for each die a normalized histogram.
\part Calculate the mean and the standard deviation for each face
value averaged over the dice.
\part Visualize the result in a bar plot with error bars
(\code{bar()} and \code{errorbar()} functions).
\end{parts}
\begin{solution}
\lstinputlisting{die2.m}
\includegraphics[width=0.5\textwidth]{die2}
\end{solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -209,6 +193,102 @@ Now we analyze several dice at once.
\end{solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Probabilities of a normal distribution}
Which fraction of a normally distributed data set is contained in ranges
that are symmetric around the mean?
\begin{parts}
\part Generate a data set $X = (x_1, x_2, ... x_n)$ of
$n=10000$ normally distributed numbers with mean $\mu=0$ and
standard deviation $\sigma=1$ (\code{randn() function}).
% \part Estimate and plot the probability density of this data set (normalized histogram).
% For a comparison plot the normal distribution
% \[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]
% into the same plot.
\part \label{onesigma} How many data values are at maximum one standard deviation
away from the mean?\\
That is, how many data values $x_i$ have the value $-\sigma < x_i < +\sigma$?\\
Compute the probability $P_{\pm\sigma}$ to get a value in this range
by counting how many data points fall into this range.
\part \label{probintegral} Compute the probability of
$-\sigma < x_i < +\sigma$ by numerically integrating over the
probability density of the normal distribution
\[ P_{\pm\sigma}=\int_{x=\mu-\sigma}^{x=\mu+\sigma} p_g(x) \, dx \; .\]
First check whether
\[ \int_{-\infty}^{+\infty} p_g(x) \, dx = 1 \; . \]
Why is this the case?
\part What fraction of the data is contained in the intervals $\pm 2\sigma$
and $\pm 3\sigma$?
Compare the results with the corresponding integrals over the normal
distribution.
\part \label{givenfraction} Find out which intervals, that are
symmetric with respect to the mean, contain 50\,\%, 90\,\%, 95\,\% and 99\,\%
of the data by means of numeric integration of the normal
distribution.
% \part \extra Modify the code of questions \pref{onesigma} -- \pref{givenfraction} such
% that it works for data sets with arbitrary mean and arbitrary standard deviation.\\
% Check your code with different sets of random numbers.\\
% How do you generate random numbers of a given mean and standard
% deviation using the \code{randn()} function?
\end{parts}
\begin{solution}
\lstinputlisting{normprobs.m}
\end{solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Central limit theorem}
According to the central limit theorem the sum of independent and
identically distributed (i.i.d.) random variables converges towards a
normal distribution, although the distribution of the randmon
variables might not be normally distributed.
With the following questions we want to illustrate the central limit theorem.
\begin{parts}
\part Before you continue reading, try to figure out yourself what
the central limit theorem means and what you would need to do for
illustrating this theorem.
\part Draw 10000 random numbers that are uniformly distributed between 0 and 1
(\code{rand} function).
\part Plot their probability density (normalized histogram).
\part Draw another set of 10000 uniformly distributed random numbers
and add them to the first set of numbers.
\part Plot the probability density of the summed up random numbers.
\part Repeat steps (d) and (e) many times.
\part Compare in a plot the probability density of the summed up
numbers with the normal distribution
\[ p_g(x) =
\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]
with mean $\mu$ and standard deviation $\sigma$ of the summed up random numbers.
\part How do the mean and the standard deviation change with the
number of summed up data sets?
\part \extra Check the central limit theorem in the same way using
exponentially distributed random numbers (\code{rande} function).
\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}
\end{questions}
\end{document}

19
statistics/exercises/exercises02.tex
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@@ -87,6 +87,23 @@ jan.benda@uni-tuebingen.de}
\begin{questions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Probabilities of a die II}
Now we analyze several dice at once.
\begin{parts}
\part Simulate 20 dice, each of which is rolled 100 times
(each die is simulated with the same random number generator).
\part Compute for this data set for each die a normalized histogram.
\part Calculate the mean and the standard deviation for each face
value averaged over the dice.
\part Visualize the result in a bar plot with error bars
(\code{bar()} and \code{errorbar()} functions).
\end{parts}
\begin{solution}
\lstinputlisting{die2.m}
\includegraphics[width=0.5\textwidth]{die2}
\end{solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question \qt{Probabilities of a normal distribution}
Which fraction of a normally distributed data set is contained in ranges
@@ -94,7 +111,7 @@ that are symmetric around the mean?
\begin{parts}
\part Generate a data set $X = (x_1, x_2, ... x_n)$ of
$n=10000$ normally distributed numbers with mean $\mu=0$ and
standard deviation $\sigma=1$ (\code{randn() Funktion}).
standard deviation $\sigma=1$ (\code{randn() function}).
\part Estimate and plot the probability density of this data set (normalized histogram).
For a comparison plot the normal distribution
\[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]

8
statistics/lecture/statistics.tex
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@@ -214,10 +214,10 @@ category $i$, i.e. of getting a data value in the $i$-th bin.
\end{exercise}
\begin{exercise}{diehistograms.m}{}
Plotte Histogramme von 20, 100, und 1000-mal W\"urfeln. Benutze
\code[hist()]{hist(x)}, erzwinge sechs Bins mit
\code[hist()]{hist(x,6)}, oder setze selbst sinnvolle Bins. Normiere
anschliessend das Histogram.
Plot histograms for 20, 100, and 1000 times rolling a die. Use
\code[hist()]{hist(x)}, enforce six bins with
\code[hist()]{hist(x,6)}, or set useful bins yourself. Normalize the
histograms appropriately.
\end{exercise}