20.

(a) The fire alarm in Mill Lane is set off at random times. The probability of an alarm during the time interval (u, u +h) is λ(u)h +o(h), where the ‘intensity function’ λ(u) may vary with the time u. Let N(t ) be the number of alarms by time t , and set N(0) = 0. Show, subject to reasonable extra conditions to be stated clearly, that pi (t) = P(N(t) = i )

satisfies p′i (t) = −λ(t )pi (t), p′i (t ) = λ(t )



pi−1(t ) − pi (t )



for i ≥ 1.