214 lines
7.0 KiB
TeX
214 lines
7.0 KiB
TeX
\documentclass[12pt,a4paper,pdftex]{exam}
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\usepackage[english]{babel}
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\usepackage{pslatex}
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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\usepackage{xcolor}
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\usepackage{graphicx}
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\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
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%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
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\pagestyle{headandfoot}
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\ifprintanswers
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\newcommand{\stitle}{: Solutions}
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\else
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\newcommand{\stitle}{}
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\fi
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\header{{\bfseries\large Exercise 6\stitle}}{{\bfseries\large Statistics}}{{\bfseries\large November 14th, 2017}}
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\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
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jan.benda@uni-tuebingen.de}
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\runningfooter{}{\thepage}{}
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\setlength{\baselineskip}{15pt}
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\setlength{\parindent}{0.0cm}
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\setlength{\parskip}{0.3cm}
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\renewcommand{\baselinestretch}{1.15}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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language=Matlab,
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basicstyle=\ttfamily\footnotesize,
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numbers=left,
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numberstyle=\tiny,
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title=\lstname,
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showstringspaces=false,
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commentstyle=\itshape\color{darkgray},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{bm}
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\usepackage{dsfont}
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\newcommand{\naZ}{\mathds{N}}
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\newcommand{\gaZ}{\mathds{Z}}
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\newcommand{\raZ}{\mathds{Q}}
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\newcommand{\reZ}{\mathds{R}}
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\newcommand{\reZp}{\mathds{R^+}}
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\newcommand{\reZpN}{\mathds{R^+_0}}
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\newcommand{\koZ}{\mathds{C}}
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%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\continue}{\ifprintanswers%
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\continuepage}{\ifprintanswers%
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\newpage
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\newsolutionpage}{\ifprintanswers%
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\newpage%
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\else
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\fi}
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%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qt}[1]{\textbf{#1}\\}
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\newcommand{\pref}[1]{(\ref{#1})}
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\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
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\newcommand{\code}[1]{\texttt{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{instructions}
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\ifprintanswers%
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\else
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\begin{itemize}
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\item Convince yourself that each single line of your code really does
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what it should do! Test it with small examples directly in the
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command line.
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\item Always try to break down your solution into small and meaningful
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functions. As soon something similar is computed more than once you
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should definitely put it into a function.
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\item Initially test computationally expensive \code{for} loops, vectors,
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matrices, etc. with small numbers of repetitions and/or
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sizes. Once it is working use large repetitions and/or sizes for
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getting a good statistics.
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\item Use the help functions of \code{matlab} (\code{help command} or
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\code{doc command}) and the internet to figure out how specific
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\code{matlab} functions are used and what features they offer. In
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addition, the internet offers a lot of material and suggestions for
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any question you have regarding your code !
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\item Please upload your solution to the exercises to ILIAS as a zip-archive with the name
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``probabilities\_\{last name\}\_\{first name\}.zip''.
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\end{itemize}
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\fi
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\begin{questions}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \textbf{Read chapter 4 of the script on ``programming style''!}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Probabilities of a die I}
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The computer can roll dice with more than 6 faces!
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\begin{parts}
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\part Simulate 10000 times rolling a die with eight faces by
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generating integer random numbers $x_i = 1, 2, \ldots 8$ .
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\part Compute the probability $P(5)$ of getting a five by counting the number of fives
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occurring in the data set.
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Does the result fit to your expectation?
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Check the probabilities $P(x_i)$ of the other numbers.
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Is the die a fair die?
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\part Store the computed probabilities $P(x_i)$ in a vector and use
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the \code{bar()} function for plotting the probabilities as a
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function of the corresponding face values.
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\part Compute a normalized histogram of the face values by means of
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the \code{hist()} and \code{bar()} functions.
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\part \extra Simulate a loaded die with the six showing up
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three-times as often as the other numbers.
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Compute a normalized histogram of the face values from rolling the loaded die 10000 times.
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\end{parts}
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\begin{solution}
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\lstinputlisting{rollthedie.m}
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\lstinputlisting{diehist.m}
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\lstinputlisting{die1.m}
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\includegraphics[width=1\textwidth]{die1}
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\end{solution}
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\continue
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Probabilities of a die II}
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Now we analyze several dice at once.
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\begin{parts}
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\part Simulate 20 dice, each of which is rolled 100 times
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(each die is simulated with the same random number generator).
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\part Compute for this data set for each die a normalized histogram.
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\part Calculate the mean and the standard deviation for each face
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value averaged over the dice.
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\part Visualize the result in a bar plot with error bars
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(\code{bar()} and \code{errorbar()} functions).
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\end{parts}
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\begin{solution}
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\lstinputlisting{die2.m}
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\includegraphics[width=0.5\textwidth]{die2}
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\end{solution}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Histogram of the normal distribution}
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\vspace{-3ex}
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\begin{parts}
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\part Generate a data set $X = (x_1, x_2, ... x_n)$ of
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$n=10000$ normally distributed random numbers with mean $\mu=0$ and
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standard deviation $\sigma=1$ (\code{randn()} function).
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\part Compute from this data set the probability $P(0\le x<0.5)$.
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\part What happens to the probability of drawing a number from a
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specific range (z.B. $P(0\le x<a)$), if this range gets smaller and
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smaller, i.e. $a \to 0$?
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Write a script that illustrates this by plotting $P(0\le x<a)$
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as a function of $a$ (use $0 \le a \le 4$).
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\part \label{manualpdf} Compute and plot the probability density of
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the data set (the normalized histogram). First, define the positions
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of the bins (width of 0.5) in a vector. Count in a \code{for} loop
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for each bin die number of data values falling into the
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bin. Finally, normalize the resulting histogram and plot it using
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the \code{bar()} function.
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\part \label{gaussianpdf} Draw into the same plot the normal
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distribution
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\[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]
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for a comparison.
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\part Plot the probability density as in (\ref{manualpdf}) and
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(\ref{gaussianpdf}), but this time by means of the \code{hist()} and
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\code{bar()} functions.
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\end{parts}
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\begin{solution}
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\lstinputlisting{normhist.m}
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\includegraphics[width=1\textwidth]{normhist}
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\end{solution}
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\end{questions}
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\end{document} |