  Summarizing results from Joseph Mecke's last fragmentary manuscripts, the
generating function and the Laplace transform for nonnegative random variables
are considered. The concept of thickening of a random variable, as an inverse
operation to thinning (which is usually applied to point processes) is
introduced, based on generating functions, and a characterization of thickable
random variables is given. Further, some new relations between exponential
distributions and their interpretation in terms of Poisson point processes are
derived with the help of the Laplace transform.
