deletions | additions       diff --git a/Consider_how_the_contours_behave__.tex b/Consider_how_the_contours_behave__.tex  index 5d2e179..c25d61b 100644  --- a/Consider_how_the_contours_behave__.tex  +++ b/Consider_how_the_contours_behave__.tex 
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Consider how the contours behave for fixed $P_{\rm obs}$ and $P_{\rm act}$ increasing from its minimum to maximum value. The probability increases for increasing $P_{\rm act}$, indicating a signal with $P_{\rm obs}$ has a increasing probability to have originated from a more distant devil with a larger $P_{\rm act}$. Conversely, for fixed $P_{\rm act}$ and decreasing $P_{\rm obs}$, the probability increases, reflecting the same bias.  Given a number density for the distribution of $P_{\rm act}$-values, $n(P_{\rm act})$, we can use the combined bias and distortion expression to calculate the resulting distribution of $P_{\rm obs}$-values, $n(P_{\rm obs})$:  \begin{equation} \begin{equation}\label{eqn:n-Pobs_from_n-Pact}  n(P_{\rm obs}) = \left( \dfrac{P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right) P_{\rm obs}^{-2} \int_{P_{\rm act} = P_{\rm obs}}^{P_{\rm max}} n(P_{\rm act}) P_{\rm act}\ dP_{\rm act}.  \end{equation}  \label{eqn:n_Pobs_from_Pact}  The integral extends between $P_{\rm obs}$ and $P_{\rm max}$ since only devils with $P_{\rm act}$ in that range can contribute. Consider, for example, a uniform distribution $n(P_{\rm act}) = k = {\rm const.}$, which gives   \begin{equation} \begin{equation}\label{eqn:n-Pobs_from_uniform_n-Pact}  n(P_{\rm obs}) = \frac{1}{2} k\ \left( \dfrac{P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right) \left[ \left( \dfrac{P_{\rm max}}{P_{\rm obs}} \right)^2 - 1\right].  \end{equation}  \label{eqn:n-Pobs_from_uniform_n-Pact}  Figure \ref{fig:n-Pobs_from_uniform_n-Pact} compares the two distributions.     
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