summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex')
-rw-r--r--Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex42
1 files changed, 42 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex b/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex
new file mode 100644
index 00000000000..554de298e9d
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex
@@ -0,0 +1,42 @@
+\documentclass[compose]{exam-n}
+\begin{document}
+
+\begin{question}{30} \comment{by Graham Woan}
+Distinguish between frequentist and Bayesian definitions of
+probability, and explain carefully how parameter estimation is
+performed in each regime.\partmarks{10}
+
+A square ccd with $M\times M$ pixels takes a dark frame for
+calibration purposes, registering a small number of electrons in
+each pixel from thermal noise. The probability of there being $n_i$
+electrons in the $i$th pixel follows a Poisson distribution, i.e.
+\begin{equation*}
+ P(n_i|\lambda) = \exp(-\lambda)\lambda^{n_i}/n_i!,
+\end{equation*}
+where $\lambda$ is the same constant for all pixels. Show that the
+expectation value of is $\langle n_i \rangle = \lambda$.
+\partmarks{5} [You may assume the relation $\sum_0^\infty \frac{x^n}{n!}=\exp(x)$.]
+
+Show similarly that
+\begin{equation*}
+ \langle n_i(n_i-1) \rangle = \lambda^2.
+\end{equation*}
+and hence, or otherwise, that the variance of $n_i$ is also
+$\lambda$.
+\partmarks{5}
+
+The pixels values are summed in columns. Show that these sums,
+$S_j$, will be drawn from a parent probability distribution that is
+approximately
+\begin{equation*} p(S_j|\lambda)=\frac{1}{\sqrt{2\pi
+M\lambda}}\exp\left[-\frac{(S_j-M\lambda)^2}{2M\lambda}\right],
+\end{equation*}
+clearly stating any theorems you use.
+\partmarks{5}
+
+Given the set of $M$ values $\{S_j\}$, and interpreting the above
+as a Bayesian likelihood, express the posterior probability for
+$\lambda$, justifying any assumptions you make.
+\partmarks{5}
+\end{question}
+\end{document}