On Dec 22, 8:14 am, "porky_pig...@my-deja.com" <porky_pig...@my- deja.com> wrote: > On Friday, December 21, 2012 11:41:44 PM UTC-5, Bill Taylor wrote: > > On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote: > Dirac delta, infinitesimals, irrational numbers, transfinite ordinals, > ... are legitimate not because they have been rigorously defined Yes, that is PRECISELY why they are legitimate. > No one has ever anywhere actually used the concept of a uniform > distributions on N to solve any problem. Sure they have. You can use it to calculate the probability that two randomly chosen naturals will be co-prime, for example. And many others of that type. -- Blunderbuss Bill ** Dogma is a bitch! (pun intended) > > So, you're saying there *exists* the uniform distribution of positive integers (or natural numbers if you wish). Well, well, well, would you please then enlighten the unwashed masses like myself and tell us that's the probability of selecting an arbitrary positive integers? > > Regards, > > PPJr.
How much time you got?
Obviously enough while we model Dirac's delta with real functions, as it is rigorous in the framework we have established for standard real analysis, there isn't the extant framework for a rigorous treatment of a non-standard probability with a uniform distribution over the naturals. So, where the goal is to not invent a brand new standard with mutually consistent results, the distribution is built from the standard as described above, modeling the asymptotics of real functions. Then, where there's instead the notion that these things exist and are concrete, instead there are currently "alternative" foundations, that are true as they reflect the true character of these objects.
As above, on the one hand there are the simple discrete uniform distributions of 0 through n, or conveniently 1 through n, or 0 through n-1, the probability of each value is 1/n. For n = N, and the set is an ordinal, 1/N is not a standard real value, but it would be somewhere between zero and one. (And yes, I know that lim_n->oo 1/n = 0.) Then, each integer has an infinitesimal probability that is a constant, their sum is one. EF is the CDF of that.
Then, another notion is that a continuous probability distribution, and its CDF, would have that said CDF ranges from zero to one, for the domain of the naturals, and to be uniform, that the difference between any two consecutive values, naturally ordered by the domain, is a constant, as it is. EF is that.
Then, another would have that a probability function for a continuous distribution, would have area one, and 1 >= p(n) >= 0 for each value. EF is that.
That is about, on the one hand, seeing the values of EF as constant monotone increasing, it is the CDF, as constant monotone non- increasing, it is the p.d.f. EF is that.
Here, constant monotone is a stronger condition than monotone.
Then, as well this distribution would have that its CDF (or CDFs, where there are distributions for each of the continuous and discrete) would, among other things, be uniformly dense in values of its range throughout [0,1]. This has various consequences for results of density of series in the reals, and as well to reconcile with standard modern mathematics, would benefit from the establishment of true foundations, for these things as they would exist. There would be a relevant construction with these properties. EF is that.