Distribution of a logit transform of a normal variable - Cross Validated

Logistic normal distribution.

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. $N(\mu,\sigma^2)$, you seek the transformation: 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: 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.