Date: Dec 20, 2012 1:12 AM
Subject: Continuous and discrete uniform distributions of N

In the consideration of what a uniform probability distribution over
the natural integers would be, we can begin as modeling it with real
functions, basically as the Dirac delta defined not just at zero, but
at each natural integer, scaled or divided by the number of integers,
with commensurate width, preserving the properties of a p.d.f. that
it's everywhere greater than or equal to zero and less than or equal
to one, and that its area or integral evaluates to one.

Then, where Dirac's delta is described as a spike to infinity at zero
with unit area, then to have a spike to one at each natural integer,
basically dividing the spike fairly among the natural integers, the
consideration then is in as to whether that still would have area one,
besides that each of the points would have f(x)=1. (Of course this is
all modeled standardly with real functions.) Yet, it has area two
(exactly because the integral of EF = 1).

Another notion of what would be the nearest analog to a uniform
probability distribution over the entire set of naturals would be for
each to have probability 1/omega, that as a discrete instead of
continuous distribution, the sum over them equals one. Here then
there's a consideration that there is a continuous distribution, of N,
because a p.d.f. exists and a p.d.f. (or CDF) defines a continuous
distribution. Then there's also a consideration that there's a
discrete distribution, of N, defined as one iota for each.

EF: continuous uniform distribution of N
(EF + REF)/2: continuous uniform distribution of N
f(x)=iota: discrete uniform distribution of N


Ross Finlayson