While analyzing data regarding the pore size distribution of a disordered material (see for example S. Bhattacharya and K.E. Gubbins, Langmuir 2006 22 18 7726-7731), I found a probability density (PDF) function which at first sight resembles a Gaussian: however, taking a closer look at the tails I noticed that the left-side tail seems to decay exponentially.
This is the rescaled PDF $P(y)$, with $y=x/\langle x \rangle-1$, in linear scale:
The red curve is a Gaussian fit. As you can see, they look quite similar on the surface.
However, if I plot $P(y)$ in semi-logarithmic scale, we can see that the left-side tail is not Gaussian at all:
The black dashed line is a fit with an exponential function. As you can see, the left-side tail looks exponential, while the right-side one looks Gaussian (even though the statistics is a bit worse there). Regarding the little bump on the left, I have no idea whether it is real or just an artifact of the analysis, so I will ignore it for the moment.
At first I thought this could be a Gumbel distribution, but this is clearly not the case since the Gumbel has a rather large skewness of $\simeq 1.14$, while the skewness of this PDF is around $-0.16$ (I also tried fitting, just to be sure: doesn't work).
Does someone know about a PDF of this kind, i.e., with an exponential left-side tail and a Gaussian-looking right side tail?