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 />
5.1 Hyper-real X()<br />
ζ<br />
X()<br />
ζ is Hyper-real <strong>R<strong>and</strong>om</strong> Variable iff<br />
its values may<br />
include infinitesimals, <strong>and</strong> infinite hyper-reals.<br />
5.2 Hyper-real Probability Distribution <strong>of</strong> X()<br />
ζ<br />
Let<br />
X()<br />
ζ<br />
be Hyper-real, <strong>and</strong> define,<br />
dF( x) = Pr( x − dx ≤ X( ζ) ≤ x + dx)<br />
.<br />
1 1<br />
2 2<br />
Then,<br />
Fx ( ) = ∑ dFx ( ).<br />
x= X( ζ), ζ∈S<br />
is a Hyper-real Probability Distribution <strong>of</strong> X()<br />
ζ<br />
Example<br />
If a ball is drawn from a container that has 5 Red balls, <strong>and</strong><br />
4 Black balls, <strong>and</strong> X()<br />
ζ is the number <strong>of</strong> Red balls,<br />
dF(0) = Pr( X( ζ) = 0) =<br />
4<br />
9<br />
dF(1) = Pr( X( ζ) = 1) = .<br />
5<br />
9<br />
20
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