Question

In this problem, we develop an alternative derivation for the mean function of the shot noise process described in Section 8.7,
Where the are the arrival times of a Poisson process with arrival rate, λ, and h (t) is an arbitrary pulse shape which we take to be causal. That is, h (t) = 0 for t < 0. In order to find the mean function, µX (t) = E [X (t)], we condition on the event that there were exactly n arrivals in [0, t]. Then, the conditional mean function is
(a) Use the results of Exercise 8.40 to justify that
Where the Xi are a sequence of IID random variables uniformly distributed over [0, t].
(b) Show that the expectation in part (a) reduces to
(c) Finally, average over the Poisson distribution of the number of arrivals to show that

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