4.8. Electrical pulses with independent and identically distributed random amplitudes 6 62, . . . arrive at a detector at random times W,, W, ... according to a Poisson process of rate A. The detector output 0k(t)

for the kth pulse at time t is 0k(t) =

0 fort < Wk, k exp{-a(t - W)} fort ? W.

That is, the amplitude impressed on the detector when the pulse arrives is ek, and its effect thereafter decays exponentially at rate

a. Assume that the detector is additive, so that if N(t) pulses arrive during the time interval

[0, t], then the output at time t is N(t)

Z(t) _ 7 Ok(t).

k=l Determine the mean output E[Z(t)] assuming N(0) = 0. Assume that the amplitudes 4, 62, . . . are independent of the arrival times W,, W . ... .