Statistics Homework Solutions
Problem
#11599

Expected value in a Poisson process

Expected value in a Poisson process
Don't know how to compute the expected value (see attached file).

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004.pdf
Suppose that electrical pulses having random amplitudes arrive at an electronic counter in
accordance with a Poisson process with rate . The amplitude of a pulse is assumed to decrease
with time at an exponential rate. That is, if a pulse has amplitude A upon arrival, then its amplitude
at a time t units later will be Ae -t , with being a constant. We further suppose that the initial
amplitude of the pulses are independent and have a common distribution F.
Let S1 , S2 , ..., be the arrival times of the pulses and let A1 , A2 , ..., be their respective amplitudes.
Then,

N (t )
B(t ) = Ae i
- (t -S i )

i =1


represents the total amplitude of the electric counter at time t. Compute the expected value E[B(t)]
of B(t). [Hint: B(t) contains two types of randomness: one due to N(t) and one due to A. It might
help to condition on N(t).]
Solution
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Amrit Lal Ahuja, PhD (IP) - 4.9/5
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