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Brian Jackson edited Consider_how_the_contours_behave__.tex
almost 9 years ago
Commit id: 3706e7a1ed0258695442e2d1ac137dad0eaf7416
deletions | additions
diff --git a/Consider_how_the_contours_behave__.tex b/Consider_how_the_contours_behave__.tex
index dafb192..b12ca71 100644
--- a/Consider_how_the_contours_behave__.tex
+++ b/Consider_how_the_contours_behave__.tex
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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}.
\label{eqn:n_Pobs_from_Pact}
\end{equation}
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
give gives rise to $n(P_{\rm obs}) = \frac{1}{2} k\ \left( \dfrac{P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right) \left[ \left( \dfrac{P_{\rm max}}{P_{\rm obs}} \right)^2 - 1\right]$. Figure \ref{fig:n-Pobs_from_uniform_n-Pact} compares the two distributions.