Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 9F / a / Solution

A geometric distribution with parameter $p$ on $\{1,2,\dots \}$ has probability mass functionIt counts the trial on which the first success occurs in independent trials with success probability $p$. Its probability generating function is $ps/[1-(1-p)s]$. Differentiating at $s=1$ givesThis support convention is important: a count of failures before the first success would start at zero.