68. Suppose that electrical shocks having random amplitudes occur at times distributed according to a Poisson process {N(t),t 0} with rate λ. Suppose that the amplitudes of the successive shocks are independent both of other amplitudes and of the arrival times of shocks, and also that the amplitudes have distribution F with mean μ. Suppose also that the amplitude of a shock decreases with time at an exponential rate α, meaning that an initial amplitude A will have value Ae−αx after an additional time x has elapsed. Let A(t) denote the sum of all amplitudes at time t. That is, A(t) =

N(t)



i=1 Aie−α(t−Si)

where Ai and Si are the initial amplitude and the arrival time of shock i.

(a) Find E[A(t)] by conditioning on N(t).

(b) Without any computations, explain why A(t) has the same distribution as does D(t) of Example 5.21.