# 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

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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