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diff --git a/macros/latex/contrib/exam-n/sample/numerical1.tex b/macros/latex/contrib/exam-n/sample/numerical1.tex new file mode 100644 index 0000000000..98157ebf68 --- /dev/null +++ b/macros/latex/contrib/exam-n/sample/numerical1.tex @@ -0,0 +1,58 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{20} \author{Graham Woan} + +\part The recently-launched \emph{Swift} Gamma Ray Burst telescope is +expected to detect about 200 bursts of gamma rays during its +2-year lifespan. Explain why the Poisson distribution, +\begin{equation*} +P(n|\lambda)=\exp(-\lambda)\lambda^n /n! +\end{equation*} +is appropriate to describe the probability of detecting $n$ +bursts, and carefully explain the significance of the parameter +$\lambda$. Table~\ref{t:excess} has absolutely nothing to do with +this question, and its presence here is proof positive of the existence of +aliens who wish to do us typographical harm.\partmarks{4} + +% This table is here to check that the table in the _solution_ to the +% previous question doesn't cause the table counter to be +% incremented. This should be Table 2, whether or not solutions are +% being shown. +\begin{table} +\begin{centering} +\begin{tabular}{r|l} +\hline +left&right\\ +\hline +\end{tabular} +\caption{\label{t:excess}This is a table} +\end{centering} +\end{table} + +Given the above, estimate the probability that \emph{Swift} will +detect more than three bursts on any particular calendar day. +% Fill up the line, so that we check that (default) \partmarks doesn't +% include \@partmarksspace, since (default) \partmarks is now fully in +% the margin. +Blah. Blah. Blaah. Fill the line.\partmarks{6} + +\part Explain how Bayesian inference uses the observed number of +bursts to infer the true burst rate at the sensitivity limit of +\emph{Swift}, and explain the significance of the posterior +probability distribution for $\lambda$. \partmarks{5} +\begin{solution} +\includepdf[pages=-]{numerical1-solution.pdf} +\end{solution} + +Assuming that the posterior, $p$, for $\lambda$ can be +approximated as a gaussian, show that, quite generally, the +uncertainty in $\lambda$ inferred from \emph{Swift} will be +\begin{equation*} +\sigma \simeq \left( -\frac{\partial^2\ln p}{\partial +\lambda^2}\Big|_{\lambda_0} \right)^{-1/2}, +\end{equation*} +where $\lambda_0$ is the most probable value of $\lambda$. +\partmarks{5} +\end{question} +\end{document} |