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Brian Jackson edited Consider_how_the_contours_behave__.tex
almost 9 years ago
Commit id: d7ac26d3c28832ba991f77fc2424d95f84c29bf7
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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}
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} %\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 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.