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correct some notational blunders in the integration chapter

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hiro98 2020-06-21 21:26:13 +02:00
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commit 48248d980a
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latex/tex/monte-carlo/integration.tex
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@ -45,7 +45,7 @@ expected value \(\EX{X_i}=\mathbb{E}\) and variance
% %
Because \(\frac{\sigma^2}{N}\xrightarrow{N\rightarrow\infty} 0\) Because \(\frac{\sigma^2}{N}\xrightarrow{N\rightarrow\infty} 0\)
\cref{eq:approxexp} really converges to \(I\). For finite, but large \cref{eq:approxexp} really converges to \(I\). For finite, but large
\(N\) the value of \cref{eq:approxexp} is distributed (in increasingly \(N\) the value of \(\EX{F/P}\) is distributed (in increasingly
good approximation) according to a normal distribution around \(I\) good approximation) according to a normal distribution around \(I\)
with the variance \(\VAR{F/\Rho}\cdot N^{-1}\) as in \cref{eq:varI} by with the variance \(\VAR{F/\Rho}\cdot N^{-1}\) as in \cref{eq:varI} by
virtue of the central limit theorem. virtue of the central limit theorem.
@ -121,7 +121,7 @@ squared when approximating the integral by the sum.
% %
\begin{equation} \begin{equation}
\label{eq:approxvar-I} \label{eq:approxvar-I}
\VAR{I} = \abs{\Omega}\int_\Omega f(\vb{x})^2 - \VAR{\frac{F}{P}} = \abs{\Omega}\int_\Omega f(\vb{x})^2 -
\underbrace{\qty(\frac{I}{\abs{\Omega}})^2}_{=\bar{f}} \dd{\vb{x}} \equiv \abs{\Omega}\cdot\sigma_f^2 \approx \underbrace{\qty(\frac{I}{\abs{\Omega}})^2}_{=\bar{f}} \dd{\vb{x}} \equiv \abs{\Omega}\cdot\sigma_f^2 \approx
\frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i) - \bar{f}]^2 \frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i) - \bar{f}]^2
\end{equation} \end{equation}
@ -129,8 +129,9 @@ squared when approximating the integral by the sum.
Applying this method to integrate the \(\qqgg\) cross section from Applying this method to integrate the \(\qqgg\) cross section from
\cref{eq:crossec} over a \(\theta\) interval, equivalent to \cref{eq:crossec} over a \(\theta\) interval, equivalent to
\(\eta\in [-2.5, 2.5]\) with a target accuracy of \(\eta\in [-2.5, 2.5]\) with a target accuracy of
\(\varepsilon=10^{-3}\) results in \result{xs/python/xs_mc} with a \(\sigma=\SI{1e-3}{\pico\barn}\) results in
sample size of \result{xs/python/xs_mc_N}. \result{xs/python/xs_mc} with a sample size of
\result{xs/python/xs_mc_N}.
Changing variables and integrating \cref{eq:xs-eta} over \(\eta\) with Changing variables and integrating \cref{eq:xs-eta} over \(\eta\) with
the same target accuracy yields~\result{xs/python/xs_mc_eta} with a the same target accuracy yields~\result{xs/python/xs_mc_eta} with a