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.