# Half-Normal distribution

## Story

The Half-Normal distribution is a Normal distribution truncated to only have nonzero probability density for values greater than or equal to the location of the peak.

## Parameters

The Half-Normal distribution is parametrized by a positive scale parameter $$\sigma$$ and a location parameter $$\mu$$. In most applications, $$\mu = 0$$.

## Support

The Half-Normal distribution is supported on the set of all real numbers that are greater than or equal to $$\mu$$, that is on $$[\mu, \infty)$$.

## Probability density function

\begin{split} \begin{align} f(y;\mu, \sigma) = \left\{\begin{array}{cll} \sqrt{\frac{2}{\pi\sigma^2}}\,\mathrm{e}^{-(y-\mu)^2/2\sigma^2} & & y \ge \mu \\[1em] 0 & & \text{otherwise}. \end{array}\right. \end{align}\end{split}

## Moments

Mean: $$\displaystyle{\mu + \sqrt{\frac{2\sigma^2}{\pi}}}$$

Variance: $$\displaystyle{\left(1 - \frac{2}{\pi}\right)\sigma^2}$$

## Usage

Package

Syntax

NumPy

mu + np.abs(rg.normal(0, sigma)

SciPy

scipy.stats.halfnorm(mu, sigma)

Stan sampling

real<lower=mu> y; y ~ normal(mu, sigma)

Stan rng

real<lower=mu> y; y = mu + fabs(normal_rng(0, sigma))

## Notes

• In Stan, a Half-Normal is defined by putting a lower bound of $$\mu$$ on the variable and then using a Normal distribution with location parameter $$\mu$$.

• The Half-Normal distribution with $$\mu = 0$$ is a useful prior for nonnegative parameters that should not be too large and may be very close to zero.