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Poisson distributed quantities

As is well known, the typical application of the Poisson distribution is in counting experiments such as source activity, cross-sections, etc. The unknown parameter to be inferred is $ \lambda$. Applying the Bayes formula we get
$\displaystyle f(\lambda\,\vert\,x,{\cal P})$ $\displaystyle =$ $\displaystyle \frac{\frac{\lambda^x\,e^{-\lambda}}{x!}
\,f_\circ(\lambda)}
{\in...
...nfty\frac{\lambda^x\,e^{-\lambda}}{x!}
\,f_\circ(\lambda) \,\rm {d}\lambda}\, .$ (5.53)

Assuming5.7 $ f_\circ(\lambda)$ constant up to a certain $ \lambda_{max}\gg x$ and making the integral by parts we obtain
$\displaystyle f(\lambda\,\vert\,x,{\cal P})$ $\displaystyle =$ $\displaystyle \frac{\lambda^x\, e^{-\lambda}}{x!}$ (5.54)
$\displaystyle F(\lambda\,\vert\,x,{\cal P})$ $\displaystyle =$ $\displaystyle 1 - e^{-\lambda}\left(\sum_{n=0}^x \frac{\lambda^n}{n!}\right)\,,$ (5.55)

where the last result has been obtained by integrating ([*]) also by parts. Figure [*] shows how to build the credibility intervals, given a certain measured number of counts $ x$.
Figure: Poisson parameter $ \lambda$ inferred from an observed number $ x$ of counts.
\begin{figure}\centering\epsfig{file=dago7.eps,clip=}\end{figure}
Figure [*] shows some numerical examples.
Figure: Examples of $ f(\lambda\,\vert\,x_i)$.
\begin{figure}\centering\epsfig{file=invpois.eps,clip=}\end{figure}

$ f(\lambda)$ has the following properties.

Let us conclude with a special case: $ x=0$. As one might imagine, the inference is highly sensitive to the initial distribution. Let us assume that the experiment was planned with the hope of observing something, i.e. that it could detect a handful of events within its lifetime. With this hypothesis one may use any vague prior function not strongly peaked at zero. We have already come across a similar case in Section  [*], concerning the upper limit of the neutrino mass. There it was shown that reasonable hypotheses based on the positive attitude of the experimentalist are almost equivalent and that they give results consistent with detector performances. Let us use then the uniform distribution
$\displaystyle f(\lambda\,\vert\,x=0,{\cal P})$ $\displaystyle =$ $\displaystyle e^{-\lambda},$ (5.63)
$\displaystyle F(\lambda\,\vert\,x=0,{\cal P})$ $\displaystyle =$ $\displaystyle 1-e^{-\lambda},$ (5.64)
$\displaystyle \lambda$ $\displaystyle <$ $\displaystyle 3 \ $   at $\displaystyle 95\,\%\ $   probability$\displaystyle \,.$ (5.65)

Figure: Upper limit to $ \lambda$ having observed 0 events.
\begin{figure}\centering\epsfig{file=dago8.eps,clip=}\end{figure}

next up previous contents
Next: Uncertainty due to systematic Up: Counting experiments Previous: Binomially distributed observables   Contents
Giulio D'Agostini 2003-05-15