Hello, everyone.

I'm having some doubts with the following problem:

Let X denote the execution time of a job rounded to the nearest second. The charges are based on a linear function Y = mX + n of the execution time for suitably chosen nonnegative integers m and n. Given the PGF (probability generating function) of X, find the PGF and pmf of Y.

I will write (shorty) what I have tried. If someone could confirm my solution or give me some hint I would appreciate that.

I started with

Hence we have that . By the euclidian algorithm exist k and r<m such that and since the time is rounded to the nearest second I considered for

With this I got

And then using the sum of the geometric progression and rearranging the terms I got

p.s. sorry for my English.