Question: Suppose that electrical shocks having random amplitudes occur at times distributed according to a Poisson process \(\{N(t), t \geqslant 0\}\) with rate \(\lambda\). Suppose that the amplitudes of the successive shocks are independent both of other amplitudes an of the arrival times of shocks, and also that the amplitudes have distribution. with mean \(\mu\). Suppose also that the amplitude of a shock decreases with time an exponential rate \(\alpha\), meaning that an initial amplitude \(A\) will have value \(A{{e}^{-\alpha t}}\) after an additional time \(x\) has elapsed. Let \(A(t)\) denote the sum of all amplitude at time \(t\). That is,

where \(A_{i}\) and \(S_{i}\) are the initial amplitude and the arrival time of shock \(i\).

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

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