[statistics] shortened exercise01
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\documentclass[12pt,a4paper,pdftex]{exam}
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\documentclass[12pt,a4paper,pdftex]{exam}
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\usepackage[english]{babel}
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\usepackage{pslatex}
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@ -242,53 +242,6 @@ that are symmetric around the mean?
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\end{solution}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Central limit theorem}
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According to the central limit theorem the sum of independent and
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identically distributed (i.i.d.) random variables converges towards a
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normal distribution, although the distribution of the randmon
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variables might not be normally distributed.
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With the following questions we want to illustrate the central limit theorem.
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\begin{parts}
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\part Before you continue reading, try to figure out yourself what
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the central limit theorem means and what you would need to do for
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illustrating this theorem.
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\part Draw 10000 random numbers that are uniformly distributed between 0 and 1
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(\code{rand} function).
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\part Plot their probability density (normalized histogram).
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\part Draw another set of 10000 uniformly distributed random numbers
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and add them to the first set of numbers.
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\part Plot the probability density of the summed up random numbers.
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\part Repeat steps (d) and (e) many times.
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\part Compare in a plot the probability density of the summed up
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numbers with the normal distribution
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\[ p_g(x) =
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\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]
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with mean $\mu$ and standard deviation $\sigma$ of the summed up random numbers.
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\part How do the mean and the standard deviation change with the
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number of summed up data sets?
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\part \extra Check the central limit theorem in the same way using
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exponentially distributed random numbers (\code{rande} function).
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\end{parts}
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\begin{solution}
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\lstinputlisting{centrallimit.m}
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\includegraphics[width=0.5\textwidth]{centrallimit-hist01}
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\includegraphics[width=0.5\textwidth]{centrallimit-hist02}
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\includegraphics[width=0.5\textwidth]{centrallimit-hist03}
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\includegraphics[width=0.5\textwidth]{centrallimit-hist05}
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\includegraphics[width=0.5\textwidth]{centrallimit-samples}
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\end{solution}
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\end{questions}
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\end{document}
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