# Question: In this problem we develop an alternative derivation for the

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

**View Solution:**## Answer to relevant Questions

In this problem, we develop an alternative derivation for the mean function of the shot noise process described in Section 8.7, Where the Si are the arrival times of a Poisson process with arrival rate, λ, and h (t) is an ...Consider the random process defined in Example 8.5. The PDF, fX (x; n), and the mean function, µX [n], were found. (a) Find the joint PDF, fX1, X2 (x1, x2; n1, n2). (b) Find the autocorrelation function, RX, X (k, n) = E ...A PCM waveform has the two states + 1 and 0. Suppose the transition matrix is The initial value of the waveform is determined by the flip of a coin, with the outcome of a head corresponding to + 1 and a tail to 0. (a) What ...A communication system sends data in the form of packets of fixed length. Noise in the communication channel may cause a packet to be received incorrectly. If this happens, then the packet is retransmitted. Let the ...Demonstrate that the two generating functions defined in Equations (9.18) and (9.19) are related byPost your question