= Folded normal distribution
A folded normal distribution is the <probability distribution> of $Y=|X|$ when $X\sim N(\mu,\sigma^2)$ and $\sigma>0$. It arises when an instrument records the magnitude of a signed <measurement error> or displacement and loses the sign. This was the measurement problem motivating <The Folded Normal Distribution by F. C. Leone and others>.<The Folded Normal Distribution by F. C. Leone and others>[Abstract: unsigned measurements and estimation of the underlying normal parameters]{ref}
Let $\Phi$ denote the <cumulative distribution function> of the <standard normal distribution>. For $y\geq0$, the event $|X|\leq y$ is $-y\leq X\leq y$, so
$$
F_Y(y)=\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{-y-\mu}{\sigma}\right).
$$
For $y<0$, $F_Y(y)=0$. Differentiation adds the two preimages of a positive magnitude:
$$
f_Y(y)=\frac{e^{-(y-\mu)^2/(2\sigma^2)}+e^{-(y+\mu)^2/(2\sigma^2)}}{\sigma\sqrt{2\pi}},\qquad y>0.
$$
There is no point mass at zero. The negative half of a <normal distribution> has been reflected onto the positive half, rather than discarded and renormalized.
Splitting the <expectation> integral at zero gives
$$
\mathbb E Y=\sigma\sqrt{\frac2\pi}e^{-\mu^2/(2\sigma^2)}+\mu\left(2\Phi(\mu/\sigma)-1\right).
$$
Since $Y^2=X^2$, its <variance> is $\mu^2+\sigma^2-(\mathbb EY)^2$. At $\mu=0$ this is the <half-normal distribution>, with mean $\sigma\sqrt{2/\pi}$ and <variance> $\sigma^2(1-2/\pi)$. For an unbiased signed error with <standard deviation> one micrometre, the mean recorded magnitude is therefore about $0.798$ micrometres, despite the underlying mean being zero. A positive average magnitude is not by itself evidence for a signed instrument offset.
The parameters $\mu$ and $-\mu$ give exactly the same folded normal distribution. Sign information cannot be recovered from these magnitudes alone; even an arbitrarily large sample cannot distinguish those two signed offsets. This loss of identifiability is a consequence of the observation rule, not an error in estimating a <normal distribution>.
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