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In the next section, we apply Equation \ref{eqn:convert_from_observed_to_actual_density} to datasets from real surveys, but as an example, consider the simple observed distribution $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = k P_{\rm obs}^{-2}$. Applying Equation \ref{eqn:convert_from_observed_to_actual_density} gives a distribution of actual parameters as follows:   \begin{equation*}  \nonumber  \rho(\Gamma_{\rm act}, P_{\rm act}) = 2 k^\prime \Gamma_{\rm act}^{-11/3} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} \right)^{3/2} P_{\rm act}^{-2}. \end{equation*}  The shaded contour plot in Figure \ref{fig:integration_path} illustrates this $\rho(\Gamma_{\rm act}, P_{\rm act})$ distribution.