# Expectation and Variance of Gaussian going through Rectified Linear or Sigmoid function

When you apply a Gaussian probability densityfunction to Rectified Linear/Sigmoid function, you will get a rectified(*)/distorted Gaussian function.

(*)rectified Gaussian function is a mixture of delta and truncated gaussian as shown belown
https://en.wikipedia.org/wiki/Rectified_Gaussian_distribution

I am wondering, what is the Expectation and Variance of this rectified/distorted Gaussian?

The Expectation of distorted Gaussian from Sigmoid is shown below.
Expected value of applying the sigmoid function to a normal distribution

How about Variance? or Expectation and Variance of rectified Gaussian from Rectified Linear?

Thank you.

• It looks like what you are a calling a rectified Gaussian is $Y=\max[c,X]$, where $c$ is a constant and $X$ is Gaussian (say, mean $m$ and variance $\sigma^2>0$). You can find $E[Y]$, $E[Y^2]$, and $Var(Y)$ easily via the law of total expectation (condition on $X\geq c$ and $X<c$). – Michael Oct 11 '16 at 6:03
• So for example rectifying at $0$ for a Normal distribution $N(\mu, \sigma^2)$ you get an expectation of $\mu\,\left(1-\Phi\left( -\dfrac{\mu}{\sigma}\right)\right) + \sigma\,\phi\left(-\dfrac{\mu}{\sigma}\right)$ – Henry Oct 11 '16 at 7:14
• Thanks for advice, yes I was able to derive same equation for expectation. But Variance is very hard and I got a very long equation... too big to paste it here. – JimSD Oct 13 '16 at 10:18