Distribution of a logit transform of a normal variable If I have a normal distribution with a given mean and variance and apply a logistic transform to it, what is the mean and variance of my transformed variable? 
This seems like it has to be a well known problem, but I haven't been able to find a quick reference anywhere. 
 A: (In response to the question in the title rather than the body, on which I have nothing to add to wolfies' answer)
Wolfies has already given the distribution of the logit-normal but I thought I'd show that it's relatively simple to derive.
Let $X\sim N(\mu,\sigma^2)$ and let $Y= t(X) = 1/(1+e^{-X})$.
\begin{eqnarray}
P(Y\leq y) &=& P[t(X)\leq y]\\\
&=& P[X\leq t^{-1}(y)]\\
&=& P[X\leq \log(\frac{y}{1-y})]\\
&=& P[\frac{X-\mu}{\sigma}\leq \frac{\log(\frac{y}{1-y})-\mu}{\sigma}]\\
&=&\Phi(\frac{\log(\frac{y}{1-y})-\mu}{\sigma}), \, 0<y<1
\end{eqnarray}
and then simple differentiation to obtain the pdf
\begin{eqnarray}
f_Y(y) &=& \frac{1}{\sigma}\frac{d\log(\frac{y}{1-y})}{dy} \phi(\frac{\log(\frac{y}{1-y})-\mu}{\sigma})\\
&=& \frac{1}{\sigma}\frac{1}{y(1-y)} \phi(\frac{\log(\frac{y}{1-y})-\mu}{\sigma})
,\: 0<y<1
\end{eqnarray}
Or you could get the same thing using the formula
$f_Y(y)= f_X(t^{-1}(y)) \left|\frac{d t^{-1}(y)}{dy}\right|$ 
more directly.
A: Given $X$ ~ $N(\mu,\sigma^2)$, you seek the transformation:
$$Y = \frac{1}{1+\exp(-X)}$$
The pdf of $Y$ has domain of support on (0,1).


*

*If $\mu$ is zero, the pdf of $Y$ will be symmetrical on (0,1), so the mean, $E[Y] = \frac12$, for all $\sigma$. 

*If $\mu$ is non-zero, the pdf is skewed, and the moment calculations do not appear easy, assuming a closed-form does exist.
Nevertheless, we can quite easily find the pdf of $Y$. You can do this using the method of transformations, or one of the alternative methods. Here, I am using the mathStatica add-on to Mathematica to do the grunt work. If $X$ has pdf f, then the pdf of $Y$, say $g(y)$ is:

defined on (0,1).
Here is a plot of the pdf $g(y)$ when $\mu=0$, with various values for $\sigma$:


And here is a plot of the pdf $g(y)$ when $\sigma=1$, with various values for $\mu$:

OP wrote: This seems like it has to be a well known problem
It is a variation on the Johnson $S_B$ (bounded) system of distributions. See Johnson's 1949 paper for more details ... 


*

*Johnson, N.L. (1949), Systems of frequency curves generated by methods of translation, Biometrika, 36, 149-176.


The moments of the $S_B$ system are extremely complicated. Johnson (1949) did obtain a solution for the mean (using his formulation, which is a bit different to the above), though it did not have a closed form.
