I have a question for an application in physics. So my description will be really concrete, sorry. It's about the estimation of a systematic error from a calibration system. I have a LED with an emission with a Gaussian time distribution. For simplification let's say its mean is 0 and its sigma is 10 (ns).
This light reach detectors at various distances. Let's say the furthest one receive only n=1 photon at a time. Therefore the time error is simply defined by the emission time distribution, thus the Gaussian of 10 ns sigma.
But closer are the detector, more photons they receive, so let's say the second furthest receive n=2 photons, then the third n=3 and so on. The point is that it is the first arriving photon that give the trigger, therefore the time of light detection. So it is the first (in time) emitted in the Gaussian window and then detected that will give the time detection. In consequences, closer the detector is, sooner this calibration time will be.
I did a "simulation" generating randomly, for instance, 3 random values on a Gaussian pdf, 100000 times, keeping always the smallest value of the 3, then I put it in a histogram. I obviously obtain a skewed gaussian. My question is how can I calculate this skewed Gaussian pdf in function of n?
Thank you very much.
As given in comments, the solution is not so hard. See previous comments for details, but the developped pdf is
$f_X(t)=Gaus(t,\mu,\sigma).n.\left[1/2-1/2\operatorname{erf}\left(\frac{t-\mu}{\sigma\sqrt{2}}\right)\right]^{n-1}$