deletions | additions       diff --git a/subsection_The_P__rm_act__.tex b/subsection_The_P__rm_act__.tex  index 4368ceb..2e58dd7 100644  --- a/subsection_The_P__rm_act__.tex  +++ b/subsection_The_P__rm_act__.tex 
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\subsection*{The $P_{\rm act}$ Recovery Bias and Distortion}  Not surprisingly, the devils with the smallest $P_{\rm act}$-values are the least likely to be detected since relatively close (small $b$) encounters are required for $P_{\rm obs} > P_{\rm min}$ and such encounters are more rare. For example, $b = 0$ would be required for a devil with $P_{\rm act} = P_{\rm min}$ to be detected, an encounter geometry with a infinitesimally small probability.     The probability for a devil to be detected is proportional to the area $A$ over its pressure profile for which $P(r) \ge P_{\rm min}$, which is $A = \pi r^2(P_{\rm act}) = \pi \left( \dfrac{\Gamma_{\rm act}(P_{\rm act})}{2} \right)^2 \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} \right)$. We're interested in the way in which this bias shifts the recovered distribution $n(P_{\rm obs})$ toward larger $P_{\rm obs}$, so we will calculate the relative recovery probability by taking the ratio ratio, assuming all devils have the same $\Gamma_{\rm act} = {\rm const.}$  \begin{equation}  f f_{\left( \Gamma_{\rm act} = {\rm const.} \right)}  \equiv A(P_{\rm act})/A(P_{\rm max}) = \left(\dfrac{\Gamma_{\rm act}(P_{\rm act})}{\Gamma_{\rm act}(P_{\rm max})} \right)^2 \left(  \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right). \end{equation}  For now, we'll assume $\Gamma_{\rm act}$ is independent of $P_{\rm act}$, giving $f_{\left( \Gamma_{\rm act} = {\rm const.} \right)} = \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right)$. As expected, the probability to detect a devil with $P_{\rm act} = P_{\rm min}$ is zero. Of course, the probability for detecting a devil with $P_{\rm act} = P_{\rm max}$ is not actually unity, just the relative probability. Calculating the actual probability would require us to define the total area of the arena over which observations were made, e.g. the area of the playa where the barometer was deployed.