Spot the distribution: A difference between two normal CDFs

**cgari** · December 14th 2009, 03:23 PM

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 $\mu$ is in some interval. For clarity: let $f(x)$ be the normal pdf and $F(x)$ the normal cdf, with the usual parameters $\mu, \sigma^2$ . The distribution family I am trying to find the name for is the conditional pdf given that $\mu$ lies in some range:

$f(x \mid \mu \in [\underline \mu, \overline \mu]) = \frac{1}{\overline \mu - \underline \mu} \int_{\underline \mu}^{\overline \mu} f(x) d\mu$

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

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

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

$\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: $\frac{1}{\sqrt{\pi}} \frac{1}{\overline \mu - \underline \mu} \int_{\frac{\underline \mu -x}{\sqrt{2} \sigma }}^{\frac{\overline \mu -x}{\sqrt{2} \sigma }} e^{-t^2} \, dt$

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

Many, many thanks if you know this!

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