# Problem

This is an interview question my friend asked me since she messed up with it. To be more specific,

What is the resulting distribution when a Gaussian $$\mathcal{N}(\mu, \sigma^2)$$ is transformed by Sigmoid $$t(x)=\frac{1}{1+e^{-x}}$$.

I tried to derive and got the following \begin{aligned} \log y&=x-\log (1+e^x)\\ x&=\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{(x-\mu)^2}{2\sigma^2}) \end{aligned} But the $$\log(\cdot)$$ seems to be intractable when I plug $$x$$ in.

I also tried to simulate this process and got

It seems that the resulting distribution is still Gaussian (with some elevation and shrinking in magnitude). But I still would like to know the theoretical derivation.

Could someone help me and my friend? Thank you in advance.

• 1. Your question (specifically the quoted part) uses the same symbol ($\sigma$) for two entirely different things. This will likely lead to disaster. 2. By "passed through" I take it you mean "transformed by", i.e. if $X\sim N(\mu,\sigma^2)$ you're seeking the density of $Y=t(X)$ where $t(x) = 1/[1+\exp(-x)]$? – Glen_b -Reinstate Monica Nov 8 '19 at 4:43
• @Glen_b Sorry about the confusion. Will update the question. – Mr.Robot Nov 8 '19 at 4:47
• Do you seek methodology (how do I compute the density of a transformed random variable?) or just the answer -- it's (a well known distribution with its own wikipedia page and many questions about it on site). – Glen_b -Reinstate Monica Nov 8 '19 at 4:49
• @Glen_b I didn't know this. Could you provide a pointer to this page. Also, it would be great if you could help me with the derivation. – Mr.Robot Nov 8 '19 at 4:52
• 1. An example of a previous question is here: Distribution of a logit transform of a normal random variable 2. Wikipedia on How to transform random variables 3. Wikipedia on the resulting distribution, the logit-normal – Glen_b -Reinstate Monica Nov 8 '19 at 5:05

Hint: Let $$Y$$ be the random variable obtained by taking a sigmoid of a normal random variable, $$X$$.
This means that $$\operatorname{logit}(Y)$$ is normal. We might call this a ______ normal random variable.
Here is a another hint: If $$\log(Y)$$ is normal, we call $$Y$$ a log normal distribution. Can you apply the same naming scheme to your problem?