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 />
X(1 R , 2) = 2 ,<br />
X(1 R , 3) = 3 .<br />
The outcome <strong>of</strong> two Red balls, appears once at<br />
four times at t = 3,<br />
X(2 R ,1) = 0 ,<br />
X(2 R , 2) = 1,<br />
X(2 R , 3) = 4 .<br />
t = 2, <strong>and</strong><br />
The outcome <strong>of</strong> three Red balls, appears once at t = 3,<br />
X(3 R ,1) = 0 ,<br />
X(3 R ,2) = 0 ,<br />
X(3 R , 3) = 1 .<br />
The sample space <strong>of</strong> the process is<br />
{0 R,1 R, 2 R, 3 R } . <br />
7.1 Hyper-real X(,)<br />
ζ t<br />
A <strong>R<strong>and</strong>om</strong> Signal is Hyper-real iff the time variable t , <strong>and</strong><br />
the values <strong>of</strong><br />
hyper-reals.<br />
X(,)<br />
ζ t<br />
may include infinitesimals, <strong>and</strong> infinite<br />
7.2 Hyper-real Probability Distribution <strong>of</strong> X(,)<br />
ζ t<br />
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