 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.^[ref]

Let $\Phi$ denote the cumulative distribution function of the standard normal distribution. For $y\ge 0$, the event $|X|\le y$ is $-y\le X\le y$, soFor $y<0$, $F_Y(y)=0$. Differentiation adds the two preimages of a positive magnitude: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 givesSince $Y^2=X^2$, its variance is ${\mu }^2+{\sigma }^2-(\mathbb{E}Y{)}^2$. 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.