deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index 9ccf37b..e6e49b6 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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\label{eqn:convert_from_actual_to_observed_density}  \rho(P_{\rm obs}, \Gamma_{\rm obs}) = \int_{b = 0}^{b_{\rm max}} f\ \rho(P_{\rm act}(b), \Gamma_{\rm act}(b))\ \dfrac{2b\ db}{b_{\rm max}^2} = \int_{b = 0}^{\frac{1}{2} \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}} f\ \rho(P_{\rm act}(b), \Gamma_{\rm act}(b))\ \left( \Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2 \right)^{-1}\ 4b\ db.  \end{equation}  The following figure shows the result for a uniform distribution for underlying values, $\rho(P_{\rm act}, \Gamma_{\rm act}) = \left( P_{\rm max} - P_{\rm min} \right)^{-1}\ \left( \Gamma_{\rm max} - \Gamma_{\rm min} \right)^{-1}$.We can use the encounter geometry to model the statistical probability for $P_{\rm obs}$ and $\Gamma_{\rm obs}$ to fall within a certain range of values, given a distribution of $P_{\rm act}$- and $\Gamma_{\rm act}$-values. The probability density for passing between $b$ and $b + db$ of a devil is $dp(b) = 2 b\ db / b_{\rm max}^2 $ for $b \le b_{\rm max}$.