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

(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

x(t) = h(t-5), 71 ELXO n arrivals in [0,t)]-,)|n arivals in [0,) 2 E(X(t)ln arrivals in [0,t)] = E[h(t-X)], EL(n arrivals in [0, t)] =-[h(t-wdu . E(X(t)] = | h(t)dt.