Spot the distribution: A difference between two normal CDFs

Does anyone know if this is a known distribution with known properties? It would assist me in writing a model if I knew I were working with a known family of distributions.

My pdf is constructed out of a normal pdf conditional on knowing $\displaystyle \mu$ is in some interval. For clarity: let $\displaystyle f(x)$ be the normal pdf and $\displaystyle F(x) $ the normal cdf, with the usual parameters $\displaystyle \mu, \sigma^2$. The distribution family I am trying to find the name for is the conditional pdf given that $\displaystyle \mu$ lies in some range: $\displaystyle f(x \mid \mu \in [\underline \mu, \overline \mu]) = \frac{1}{\overline \mu - \underline \mu} \int_{\underline \mu}^{\overline \mu} f(x) d\mu $

There are a few ways to write this integral. I include all of them in case anyone has seen any of them before:

- Use $\displaystyle F(x \mid \mu , \sigma^2)$ to represent the normal CDF with parameters $\displaystyle \mu, \sigma^2$.

Then, my conditional PDF can be written: $\displaystyle \frac{F(-x \mid -\overline \mu, \sigma^2) - F(-x \mid -\underline \mu, \sigma^2)}{\overline \mu - \underline \mu} $

- $\displaystyle \frac{1}{2} \frac{1}{\overline \mu - \underline \mu} \left(\text{Erf}\left[\frac{\overline \mu -x}{\sqrt{2} \sigma }\right]-\text{Erf}\left[\frac{\underline \mu -x}{\sqrt{2} \sigma }\right] \right) \ $, where Erf is the error function.

- A third way is to write the integral explicitly: $\displaystyle \frac{1}{\sqrt{\pi}} \frac{1}{\overline \mu - \underline \mu} \int_{\frac{\underline \mu -x}{\sqrt{2} \sigma }}^{\frac{\overline \mu -x}{\sqrt{2} \sigma }} e^{-t^2} \, dt $

The pdf seems to have a number of nice properties. For example, it is symmetric and the mean is $\displaystyle \frac{\overline \mu + \underline \mu}{2} $. Its variance is $\displaystyle \sigma^2 + \frac{(\overline \mu-\underline \mu)^2}{12} $. Its graph looks a little like a generalized normal, and indeed it seems to tend toward the uniform as $\displaystyle \overline \mu - \underline \mu \rightarrow \infty $. I'm looking for guidance because I don't want to reinvent the wheel!

Many, many thanks if you know this!