Let's suppose that we have a curve that is a Gaussian PDF (probability density function) normalized to an area of one. That is a normal distribution.
If we take that gaussian but change it's std to half of what it was, will this gaussian curve have 50% the space it had, such that we'd need to double the height of the curve to renormalize to a PDF that has an area of 1 again?
If not, how to guess the ratio by which to change the std to apply a reduction of half? In other words, what is the new std for an area multiplier of 0.5, having fixed the mean and the height of the Gaussian.
So, "Is the area of a Gaussian proportional to its standard deviation (std)?"
- My guess is that the answer is yes.
The usual definition of the general Gaussian is always to be a probability density function, that is, it is defined to always have area 1. Certainly then the value of the density function at $x=0$ will shrink or grow as the standard deviation grows or shrinks.