Appendix 1—figure 1. Quality of the analytical approximation to $\stackrel{-}{\mathit{P}}$.

Upper left panel: the Monte-Carlo estimate of $\stackrel{-}{P}$ (color-coded) is shown as a function of both the mean $\mu \in \left[-\mathrm{4,4}\right]$ (y-axis) and the variance ${\sigma }^2\in \left[\mathrm{0,4}\right]$ (x-axis) of the parent process $x~N\left(\mu ,{\sigma }^2\right)$. Upper right panel: analytic approximation to $\stackrel{-}{P}$ as given by Equation A3 (same format). Lower left panel: the error, that is, the difference between the Monte-Carlo and the analytic approximation (same format). Lower right panel: the analytic approximation (y-axis) is plotted as a function of the Monte-Carlo estimate (x-axis) for each pair of moments $\left\{\mu ,{\sigma }^2\right\}$ of the parent distribution.