242 lines
11 KiB
TeX
242 lines
11 KiB
TeX
\Part[
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\eng{Probability theory}
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\ger{Wahrscheinlichkeitstheorie}
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]{pt}
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\begin{formula}{mean}
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\desc{Mean}{Expectation value}{}
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\desc[german]{Mittelwert}{Erwartungswert}{}
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\eq{\braket{x} = \int w(x)\, x\, \d x}
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\end{formula}
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\begin{formula}{variance}
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\desc{Variance}{Square of the \fqEqRef{pt:std-deviation}}{}
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\desc[german]{Varianz}{Quadrat der\fqEqRef{pt:std-deviation}}{}
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\eq{\sigma^2 = (\Delta \hat{x})^2 = \Braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
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\end{formula}
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\begin{formula}{covariance}
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\desc{Covariance}{}{}
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\desc[german]{Kovarianz}{}{}
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\eq{\cov(x,y) = \sigma(x,y) = \sigma_{XY} = \Braket{(x-\braket{x})\,(y-\braket{y})}}
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\end{formula}
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\begin{formula}{std-deviation}
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\desc{Standard deviation}{}{}
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\desc[german]{Standardabweichung}{}{}
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\eq{\sigma = \sqrt{\sigma^2} = \sqrt{(\Delta x)^2}}
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\end{formula}
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\begin{formula}{median}
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\desc{Median}{Value separating lower half from top half}{$x$ dataset with $n$ elements}
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\desc[german]{Median}{Teilt die untere von der oberen Hälfte}{$x$ Reihe mit $n$ Elementen}
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\eq{
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\textrm{med}(x) = \left\{ \begin{array}{ll} x_{(n+1)/2} & \text{$n$ \GT{odd}} \\ \frac{x_{(n/2)}+x_{((n/2)+1)}}{2} & \text{$n$ \GT{even}} \end{array} \right.
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}
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\end{formula}
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\begin{formula}{pdf}
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\desc{Probability density function}{Random variable has density $f$. The integral gives the probability of $X$ taking a value $x\in[a,b]$.}{$f$ normalized: $\int_{-\infty}^\infty f(x) \d x= 1$}
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\desc[german]{Wahrscheinlichkeitsdichtefunktion}{Zufallsvariable hat Dichte $f$. Das Integral gibt Wahrscheinlichkeit an, dass $X$ einen Wert $x\in[a,b]$ annimmt}{$f$ normalisiert $\int_{-\infty}^\infty f(x) \d x= 1$}
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\eq{P([a,b]) := \int_a^b f(x) \d x}
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\end{formula}
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\begin{formula}{cdf}
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\desc{Cumulative distribution function}{}{$f$ probability density function}
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\desc[german]{Kumulative Verteilungsfunktion}{}{$f$ Wahrscheinlichkeitsdichtefunktion}
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\eq{F(x) = \int_{-\infty}^x f(t) \d t}
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\end{formula}
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\begin{formula}{autocorrelation}
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\desc{Autocorrelation}{Correlation of $f$ to itself at an earlier point in time, $C$ is a covariance function}{}
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\desc[german]{Autokorrelation}{Korrelation vonn $f$ zu sich selbst zu einem früheren Zeitpunkt. $C$ ist auch die Kovarianzfunktion}{}
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\eq{C_A(\tau) = \lim_{T \to \infty} \frac{1}{2T}\int_{-T}^{T} f(t+\tau) f(t) \d t) = \braket{f(t+\tau)\cdot f(t)}}
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\end{formula}
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\Section[
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\eng{Distributions}
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\ger{Verteilungen}
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]{distributions}
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\Subsubsection[
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\eng{Gauß/Normal distribution}
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\ger{Gauß/Normal-Verteilung}
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]{normal}
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\begin{minipage}{\distleftwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/distribution_gauss.pdf}
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\end{figure}
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\end{minipage}
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\begin{distribution}
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\disteq{parameters}{\mu \in \R,\quad \sigma^2 \in \R}
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\disteq{support}{x \in \R}
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\disteq{pdf}{\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}
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\disteq{cdf}{\frac{1}{2}\left[1 + \erf \left(\frac{x-\mu}{\sqrt{2}\sigma}\right)\right]}
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\disteq{mean}{\mu}
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\disteq{median}{\mu}
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\disteq{variance}{\sigma^2}
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\end{distribution}
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\begin{formula}{standard_normal_distribution}
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\desc{Density function of the standard normal distribution}{$\mu = 0$, $\sigma = 1$}{}
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\desc[german]{Dichtefunktion der Standard-Normalverteilung}{$\mu = 0$, $\sigma = 1$}{}
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\eq{\varphi(x) = \frac{1}{\sqrt{2\pi}} \e^{-\frac{1}{2}x^2}}
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\end{formula}
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\Subsubsection[
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\eng{Cauchys / Lorentz distribution}
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\ger{Cauchy / Lorentz-Verteilung}
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]{cauchy}
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\begin{minipage}{\distleftwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/distribution_cauchy.pdf}
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\end{figure}
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\end{minipage}
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\begin{distribution}
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\disteq{parameters}{x_0 \in \R,\quad \gamma \in \R}
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\disteq{support}{x \in \R}
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\disteq{pdf}{\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}}
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\disteq{cdf}{\frac{1}{\pi}\arctan\left(\frac{x-x_0}{\gamma}\right) + \frac{1}{2}}
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\disteq{mean}{\text{\GT{undefined}}}
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\disteq{median}{x_0}
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\disteq{variance}{\text{\GT{undefined}}}
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\end{distribution}
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\begin{ttext}
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\eng{Also known as \textbf{Cauchy-Lorentz distribution}, \textbf{Lorentz(ian) function}, \textbf{Breit-Wigner distribution}.}
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\ger{Auch bekannt als \textbf{Cauchy-Lorentz Verteilung}, \textbf{Lorentz Funktion}, \textbf{Breit-Wigner Verteilung}.}
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\end{ttext}
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\Subsubsection[
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\eng{Binomial distribution}
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\ger{Binomialverteilung}
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]{binomial}
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\begin{ttext}
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\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \hyperref[sec:pb:distributions::poisson]{poisson distribution}}
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\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions::poisson]{Poissonverteilung}}
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\end{ttext}
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\begin{minipage}{\distleftwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/distribution_binomial.pdf}
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\end{figure}
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\end{minipage}
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\begin{distribution}
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\disteq{parameters}{n \in \Z, \quad p \in [0,1],\quad q = 1 - p}
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\disteq{support}{k \in \{0,\,1,\,\dots,\,n\}}
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\disteq{pmf}{\binom{n}{k} p^k q^{n-k}}
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% \disteq{cdf}{\text{regularized incomplete beta function}}
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\disteq{mean}{np}
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\disteq{median}{\floor{np} \text{ or } \ceil{np}}
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\disteq{variance}{npq = np(1-p)}
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\end{distribution}
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\Subsubsection[
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\eng{Poisson distribution}
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\ger{Poissonverteilung}
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]{poisson}
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\begin{minipage}{\distleftwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/distribution_poisson.pdf}
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\end{figure}
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\end{minipage}
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\begin{distribution}
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\disteq{parameters}{\lambda \in (0,\infty)}
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\disteq{support}{k \in \N}
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\disteq{pmf}{\frac{\lambda^k \e^{-\lambda}}{k!}}
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\disteq{cdf}{\e^{-\lambda} \sum_{j=0}^{\floor{k}} \frac{\lambda^j}{j!}}
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\disteq{mean}{\lambda}
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\disteq{median}{\approx\floor*{\lambda + \frac{1}{3} - \frac{1}{50\lambda}}}
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\disteq{variance}{\lambda}
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\end{distribution}
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\Subsubsection[
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\eng{Maxwell-Boltzmann distribution}
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\ger{Maxwell-Boltzmann Verteilung}
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]{maxwell-boltzmann}
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\begin{minipage}{\distleftwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/distribution_maxwell-boltzmann.pdf}
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\end{figure}
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\end{minipage}
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\begin{distribution}
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\disteq{parameters}{a > 0}
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\disteq{support}{x \in (0, \infty)}
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\disteq{pdf}{\sqrt{\frac{2}{\pi}} \frac{x^2}{a^3} \exp\left(-\frac{x^2}{2a^2}\right)}
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\disteq{cdf}{\erf \left(\frac{x}{\sqrt{2} a}\right) - \sqrt{\frac{2}{\pi}} \frac{x}{a} \exp\left({\frac{-x^2}{2a^2}}\right)}
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\disteq{mean}{2a \frac{2}{\pi}}
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\disteq{median}{}
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\disteq{variance}{\frac{a^2(3\pi-8)}{\pi}}
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\end{distribution}
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% \begin{distribution}{maxwell-boltzmann}
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% \distdesc{Maxwell-Boltzmann distribution}{}
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% \distdesc[german]{Maxwell-Boltzmann Verteilung}{}
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% \disteq{parameters}{}
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% \disteq{pdf}{}
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% \disteq{cdf}{}
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% \disteq{mean}{}
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% \disteq{median}{}
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% \disteq{variance}{}
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% \end{distribution}
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\Subsection[
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\eng{Central limit theorem}
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\ger{Zentraler Grenzwertsatz}
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]{cls}
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\begin{ttext}
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\eng{
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Suppose $X_1, X_2, \dots$ is a sequence of independent and identically distributed random variables with $\braket{X_i} = \mu$ and $(\Delta X_i)^2 = \sigma^2 < \infty$.
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As $N$ approaches infinity, the random variables $\sqrt{N}(\bar{X}_N - \mu)$ converge to a normal distribution $\mathcal{N}(0, \sigma^2)$.
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\\ That means that the variance scales with $\frac{1}{\sqrt{N}}$ and statements become accurate for large $N$.
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}
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\ger{
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Sei $X_1, X_2, \dots$ eine Reihe unabhängiger und gleichverteilter Zufallsvariablen mit $\braket{X_i} = \mu$ und $(\Delta X_i)^2 = \sigma^2 < \infty$.
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Für $N$ gegen unendlich konvergieren die Zufallsvariablen $\sqrt{N}(\bar{X}_N - \mu)$ zu einer Normalverteilung $\mathcal{N}(0, \sigma^2)$.
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\\ Das bedeutet, dass die Schwankung mit $\frac{1}{\sqrt{N}}$ wächst und Aussagen für große $N$ scharf werden.
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}
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\end{ttext}
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\Section[
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\eng{Propagation of uncertainty / error}
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\ger{Fehlerfortpflanzung}
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]{error}
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\begin{formula}{generalised}
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\desc{Generalized error propagation}{}{$V$ \fqEqRef{pt:covariance} matrix, $J$ \fqEqRef{ana:jacobi-matrix}}
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\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{pt:covariance} Matrix, $J$ \fqEqRef{ana:jacobi-matrix}}{}
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\eq{V_y = J(x) \cdot V_x \cdot J^{\T} (x)}
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\end{formula}
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\begin{formula}{uncorrelated}
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\desc{Propagation of uncorrelated errors}{Linear approximation}{}
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\desc[german]{Fortpflanzung unabhängiger fehlerbehaftete Größen}{Lineare Näherung}{}
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\eq{u_y = \sqrt{ \sum_{i} \left(\pdv{y}{x_i}\cdot u_i\right)^2}}
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\end{formula}
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\begin{formula}{weight}
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\desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fqEqRef{pt:variance}}
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\desc[german]{Gewicht}{Varianz ist eine mögliche Wahl für ein Gewicht}{}
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\eq{w_i = \frac{1}{\sigma_i^2}}
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\end{formula}
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\begin{formula}{weighted-mean}
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\desc{Weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}}
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\desc[german]{Gewichteter Mittelwert}{}{}
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\eq{\overline{x} = \frac{\sum_{i} (x_i w_i)}{\sum_i w_i}}
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\end{formula}
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\begin{formula}{weighted-mean-error}
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\desc{Variance of weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}}
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\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
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\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
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\end{formula}
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