I'm trying to resolve this exercise:
Prove that the two ways of sampling of the step length $s$ employed by Geant4 and Penelope are equivalent. In other words, demonstrate that sampling the individual step lenghts $s_i$ from $\displaystyle \mu_i e^{−\mu_i s_i}$ and then taking $s = min\{s_1 . . . s_N \}$ is the same as sampling $s$ from $\mu e^{\mu s}$ , with $\mu =\sum_i \mu_i$.
I already verified the equivalence using a code in python (that i delete from the post as suggested). I obtained this output.
Now i want to demostrate this equivalence analitically.
How can i do that? Any hint?