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


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
Regards,
Ross Finlayson

