In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval ${\displaystyle (0,\infty )}$) as a result of censoring.

The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution, derived from the normal distribution ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2}),}$ are displayed as ${\displaystyle X\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2})}$, is given by $${\displaystyle f(x;\mu ,\sigma ^{2})=\Phi {\left(-{\frac {\mu }{\sigma }}\right)}\delta (x)+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}{\textrm {U}}(x).}$$

Here, ${\displaystyle \Phi (x)}$ is the cumulative distribution function (cdf) of the standard normal distribution: $${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\quad x\in \mathbb {R} ,}$$ ${\displaystyle \delta (x)}$ is the Dirac delta function $${\displaystyle \delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}$$ and, ${\displaystyle {\textrm {U}}(x)}$ is the unit step function: $${\displaystyle {\textrm {U}}(x)={\begin{cases}0,&x\leq 0,\\1,&x>0.\end{cases}}}$$

Since the unrectified normal distribution has mean ${\displaystyle \mu }$ and since in transforming it to the rectified distribution some probability mass has been shifted to a higher value (from negative values to 0), the mean of the rectified distribution is greater than ${\displaystyle \mu .}$

Since the rectified distribution is formed by moving some of the probability mass toward the rest of the probability mass, the rectification is a mean-preserving contraction combined with a mean-changing rigid shift of the distribution, and thus the variance is decreased; therefore the variance of the rectified distribution is less than ${\displaystyle \sigma ^{2}.}$

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A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non-negative) rectified factor analysis. Harva proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory networks.