The randomly stopped geometric Brownian motion

In this short note we compute the probability density function of the random variable XTXT, where XtXt is a geometric Brownian motion, and where TT is a random variable independent of XtXt and has either a Gamma distribution or it is uniformly distributed. In the last section of the note, the distribution obtained for XTXT is fitted to the data consisting in the academic production of a set of mathematicians.