March 26, 2013

Split gaussian mathematical constant?

Assuming you have a standard Gaussian bell curve (in blue). Suppose that you want to cut it into two parts of equal areas, with an horizontal line. 

Which fraction of the Gaussian peak height provides you with the red and the green curves, which sum up to the Gaussian, with equal surface integral (undr the red and the green cuves)?

It turns out that, numerically, the fraction, on the y-axis, is about 0.3063622804625085, or one over 3.26410940175247 of the peak height.

If one looks at the x-axis, one has to cut at +/- 1.538172262286592.\sigma, where \sigma is the usual Gaussian scale parameter. In practice, cutting the Gaussian at 3/10 of the height would be good enough, assuming sufficient, far to critical, sampling. Yet out of curiosity, i looked at several numerical constant tables, or even Plouffe's constant inverter, and did not find any of these three. So once again, the potential gaussian split constants  are:
  • 0.3063622804625085
  • 1.538172262286592
  • 3.26410940175247
Does anybody knows whether this Gaussian split is "common practice" in some mathematical field, or if these constants are listed somewhere?Though application dwells in the realm of fast Gaussian filter approximation. More to come.