5.4.8 Electrical pulses with independent and identically distributed random amplitudes

1; 2; : : : arrive at a detector at random times W1;W2; : : : according to a Poisson process of rate . The detector output k.t/ for the kth pulse at time t is

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

Determine the mean output E[Z.t/] assuming N.0/ D 0. Assume that the amplitudes 1; 2; : : : are independent of the arrival times W1;W2; : : : :

Jo Ok (t)= fort < Wk, [*exp(-a(t-W)} fort Wk.