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Consider the following integral:  \begin{equation}  F = \int dx dy f P(f, P(f| \mu,  \sigma) \end{equation}  where $f$ is the number of photon counts in a pixel $(x, y)$ and $P(f, $P(f|\mu,  \sigma)$ is a Gaussian distribution centered at $\mu$ and with dispersion $\sigma$. Assuming ergodicity, we can reduce this integral to  \begin{equation}  F = \int_{-\infty}^{\infty} df f P(f, P(f| \mu,  \sigma) = \mu \end{equation}  Instead if we decided to do,  \begin{equation}  F = \int dx dy |f| P(f, P(f| \mu,  \sigma) \end{equation}  Then we would end up with  \begin{eqnarray}  F &=& 2 \int_{0}^{\infty} df f P(f, P(f| \mu,  \sigma) \\ &=& 2 \int_{0}^{\infty} df f \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{f^2}{2\sigma^2} \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(f-\mu)^2}{2\sigma^2}  \right) \\ &=& 2 \int_{0}^{\infty} \frac{\sigma dt}{\sqrt{2\pi}}\exp(-t) \\  &=& \sqrt{\frac{2}{\pi}} \sigma  \end{eqnarray}