The PDF for the half-normal distribution (HND) was missing, so put it in. Also, the physical situation in which an HND arises, is for example when magnitudes and not signed distances occur (e.g., residuals during regression), and when there is no organized bias (e.g., no bent-line residuals during regression). Now, if during regression there are points, the groupings of which, tend to be bent (i.e., when non-linear) and not arranged randomly above or below the regression line, then a folded-normal distribution would result. Thus the HND can be understood as arising in situations which are generally FND's. Now the text prior version specified that the HND is a ND folded at a (ND) mean of zero. And whereas that is absolutely true in a mathematical sense, it is also not as robust as the FND explanation. It was also confusing as after folding the (ND) mean is no longer zero (and the HND mean was not specified, well, maybe not, see below.).
Now, I would request discussion of several things here. The Mathematica reference uses theta, the reciprocal of the mean which has been left out of the development of this article. I have gotten used to theta and its advantage of not blowing up and degenerating during regression, which is, I think, why it is used. However, the reciprocal is used in the rest of the article. Should theta be propagated throughout the article? I have left it half-way, making only the minimum changes I though needed, but look forward to comments. I do not understand the Expected value given. Can someone demonstrate this to me, please? CarlWesolowski (talk) 01:40, 19 December 2011 (UTC)
In response to CarlWesolowski, the parametrization of the half-normal you added used its own mean. Specifically, it was not immediately clear, given the link to the folded-normal page, that was the mean of the half-normal and ***not*** the mean of the normal distribution being folded (which I have called X). I hope my edit has slightly clarified this (though potentially adding as a subscript would be more explicit).
Further, I have included the pdf under the alternative parametrization (to mirror the normal distribution page), and to ensure the latter parts of the article (which I have not edited) follow on logically - instead of switching parametrization part way through the page
- Neither parametrization should be preferred, since they are both considered standard in different settings (as is discussed on the normal distribution page, so perhaps a link back to that section could be added).
- So the theta should not, in my opinion, be propagated through the article.
- The derivation of the expected value follows from the integral-substitution method (as indicated in the article already).