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diff --git a/doc/sddm.tex b/doc/sddm.tex index 7da66e9..f4919e2 100644 --- a/doc/sddm.tex +++ b/doc/sddm.tex @@ -635,6 +635,25 @@ measurement and many of them end up reading outside of the linear TAC region. Therefore we also tag any events in which less than 70\% of the PMT hits have a TAC value above 400. +\appendix +\chapter{Poisson Binomial} + +Suppose we have a Poisson process whose output is then subject to a binomial +process. For example, we expect $\mu$ background events on average and we can +detect them with probability $p$. What is the probability of detecting $n$ +background events? + +\begin{align} +p(n) &= \sum_{N=n}^{\infty} P(n|N) P(N) \\ + &= \sum_{N=n}^{\infty} \frac{N!}{n!(N-n)!} p^n (1-p)^{N-n} e^{-\mu}\frac{\mu^N}{N!} \\ + &= \sum_{N=n}^{\infty} \frac{1}{n!(N-n)!} p^n (1-p)^{N-n} e^{-\mu}\mu^N \\ + &= e^{-\mu} \frac{p^n}{n!} \sum_{N=n}^{\infty} \frac{1}{(N-n)!} (1-p)^{N-n} \mu^N \\ + &= e^{-\mu} \frac{(\mu p)^n}{n!} \sum_{N=n}^{\infty} \frac{\left(\mu (1-p)\right)^{N-n}}{(N-n)!} \\ + &= e^{-\mu p} \frac{(\mu p)^n}{n!} +\end{align} + +Therefore the end result is a Poisson distribution with mean $\mu p$. + \begin{thebibliography}{9} \bibitem{grossman2017} Grossman, et al. \textit{Self-Destructing Dark Matter}. \href{https://arxiv.org/abs/1712.00455}{{\tt arXiv:1712.00455}}. Dec 2017. |