# Double random number from a gaussian, how to evaluate the skewness

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.

• Your description is quite unclear. How can emission time have mean zero? Time can't take negative values. Is emission time SPACING between consequtive photon emissions? Or do you have multiple sources all emitting one photon all together? Finding a distriubtion of a minimum of n i.i.d. Gaussians is very easy: $P(M_n>t)=(P(N>t))^n$ – A.S. Feb 18 '16 at 10:49
• Hi,sorry if it gives confusion, but it's just a convention. The 0 is the mean of the gaussian as 00:00:00 is midnight. The arrival time is obviously bigger than the emission time. – Ice Feb 18 '16 at 10:57
• Do you have multiple sources emitting simultaneously and independently or a single source emitting one photon after another? (or multiple sources emitting one photon after another? – A.S. Feb 18 '16 at 10:59
• No, it's only one source emitting many photons withing a 10 ns gaussian. and just few of them reach the detectors. – Ice Feb 18 '16 at 11:45
• OK. Then use what I wrote: $\bar F_{M_n}(t)=P(M_n>t)=(\bar F_X(t))^n$ or $f_{M_n}(t)=f_X(t)n(\bar F(t))^{n-1}$ where the last terms highlights the "skew" from pdf of you Gaussian $X\sim N(0,10ms)$. But how do you know how many photons are emitted and how many of them a receiver gets? – A.S. Feb 18 '16 at 11:50

$f_X(t)=Gaus(t,\mu,\sigma).n.\left[1/2-1/2\operatorname{erf}\left(\frac{t-\mu}{\sigma\sqrt{2}}\right)\right]^{n-1}$