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\end{equation}  Then we would end up with  \begin{equation}  F' = \int_{-\infty}^{0} df -f (-f)  P(f| \mu, \sigma) + \int_{0}^{\infty} df f P(f| \mu, \sigma) \end{equation}  \begin{equation}  \,\, = 2 \int_{0}^{\infty} df f \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(f-\mu)^2}{2\sigma^2} \right)