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 />
7.<br />
Hyper-real <strong>R<strong>and</strong>om</strong> Signal<br />
A <strong>R<strong>and</strong>om</strong> Signal (=<strong>R<strong>and</strong>om</strong> Process) is a <strong>R<strong>and</strong>om</strong><br />
Variable that depends also on the time t :<br />
X(,)<br />
ζ t .<br />
Then, the outcome <strong>of</strong> a Black ball,<br />
ζ =<br />
B<br />
is identified with the outcome <strong>of</strong> drawing one Black ball, <strong>and</strong><br />
one Red ball successively,<br />
BR , <strong>and</strong> RB ,<br />
<strong>and</strong> with the drawing <strong>of</strong> one Black ball, <strong>and</strong> two Red balls<br />
successively,<br />
etc.<br />
BRR , RBR , RRB ,<br />
For a given outcome ζ 0<br />
,<br />
X( ζ , t) = x ( t)<br />
,<br />
0<br />
is a function <strong>of</strong> t , a Sample Function, or Process Realization.<br />
Example<br />
ζ<br />
0<br />
At time<br />
t = 1, a ball is drawn from a container that has 5<br />
Red balls, <strong>and</strong> 4 Black balls, <strong>and</strong> X(,1)<br />
ζ is the number <strong>of</strong><br />
26
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