Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
13.<br />
<strong>Poisson</strong> Process<br />
The arrival at rate<br />
λ , <strong>of</strong> radioactive particles at a counter is<br />
modeled by the <strong>Poisson</strong> Process. It models other processes,<br />
such as the arrival <strong>of</strong> phone calls at rate<br />
λ , to an operator.<br />
13.1 The Bernoulli <strong>R<strong>and</strong>om</strong> Variables <strong>of</strong> the Process<br />
We assume that<br />
an arrival probability in time<br />
dt<br />
is<br />
<strong>and</strong> no arrival probability in time<br />
p<br />
= λdt<br />
,<br />
dt<br />
is<br />
q<br />
= 1 −λdt.<br />
At fixed time t , after<br />
N infinitesimal t ime intervals dt ,<br />
N<br />
=<br />
t<br />
dt<br />
, is an infinite hyper-real,<br />
there are<br />
<strong>and</strong><br />
k<br />
k arrivals,<br />
is a finite hyper-real<br />
N<br />
− k no arrivals,<br />
50
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