5. (a) A telephone answering service receives calls whose frequency varies with time but independently of other calls perhaps with a daily pattern-more during the day than the night. The rate λ(t) ≥ 0 becomes a function of the time t. The probability that a call arrives in the small time interval (t,+ ot) when n calls have been received at timer satisfies with P.(t + 1) =P-1(1)(A(t)öt+o(St)) + p,(t)(1 - A(t)st+o(dt)), (n ≥ 1), Po(+5)=(1-(1)ôt + o(öt))po(t). It is assumed that the probability of two or more calls arriving in the time interval (t,t + St) is negligible. Find the set of differential-difference equations for p,(t). Hence show that the probability generating function for the process is given by G(s,t) = exp [(x-1)(u)du] [9 marks]