Battery in a machine component needs replacement after 1, 2, or 3 time cycles, with respective probability of $p$, $2p$ and $2r$.
(a) Illustrate that the probability generating function (p.g.f.) of time to replacement of a random installed battery is given by
$F(s) = s(p + 2ps + (1 - 3p)s^2)$
(3 marks)
(b) By writing the p.g.f. of the waiting time to the $r^{th}$ replacement $\Pi w_r$ in terms of $F(s)$, illustrate that the probability of a single replacement in 2 time is $3p - p^2$.
(8 marks)
(c) Find, in terms of $p$, the probability that a battery replacement is needed in the third time cycle.
(4 marks)
