Using the \sigma parametrization of the normal distribution, the
probability density function (PDF) of the half-normal is given by : f_Y(y; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp \left( -\frac{y^2}{2\sigma^2} \right) \quad y \geq 0, where E[Y] = \mu = \frac{\sigma\sqrt{2}}{\sqrt{\pi}}. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac{\sqrt{\pi}}{\sigma\sqrt{2}}, the
probability density function is given by : f_Y(y; \theta) = \frac{2\theta}{\pi}\exp \left( -\frac{y^2\theta^2}{\pi} \right) \quad y \geq 0, where E[Y] = \mu = \frac{1}{\theta}. The
cumulative distribution function (CDF) is given by : F_Y(y; \sigma) = \int_0^y \frac{1}{\sigma}\sqrt{\frac{2}{\pi}} \, \exp \left( -\frac{x^2}{2\sigma^2} \right)\, dx Using the change-of-variables z = x/(\sqrt{2}\sigma), the CDF can be written as : F_Y(y; \sigma) = \frac{2}{\sqrt{\pi}} \,\int_0^{y/(\sqrt{2}\sigma)}\exp \left(-z^2\right)dz = \operatorname{erf}\left(\frac{y}{\sqrt{2}\sigma}\right), where erf is the
error function, a standard function in many mathematical software packages. The quantile function (or inverse CDF) is written: :Q(F;\sigma)=\sigma\sqrt{2} \operatorname{erf}^{-1}(F) where 0\le F \le 1 and \operatorname{erf}^{-1} is the
inverse error function The expectation is then given by : E[Y] = \sigma \sqrt{2/\pi}, The variance is given by : \operatorname{var}(Y) = \sigma^2\left(1 - \frac{2}{\pi}\right). Since this is proportional to the variance σ2 of
X,
σ can be seen as a
scale parameter of the new distribution. The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus, : h(Y) = \frac{1}{2} \log_2 \left( \frac{\pi e \sigma^2}{2} \right) = \frac{1}{2} \log_2 \left( 2\pi e \sigma^2 \right) -1. ==Applications==