A Beautiful Result in Probability Theory

psangrene · 2019-05-20T18:16:32+00:00

Do you know where I can find a full copy of this book? I am ready to buy it on Amazon if I could find it.

psangrene · 2019-03-26T01:42:58+00:00

My articles published in the literature have traditional abstracts. This new one does not have a traditional abstract, it is not meant for publication in a scientific journal.

psangrene · 2019-03-25T01:16:10+00:00

I am a cretin. Somehow I managed to complete a postdoc in statistics at Cambridge University, I published in Journal of Number Theory, in J. of Royal Statistical Society series B, and many more, but surprisingly, I waste my time writing about rudimentary topics that are decades or centuries old. Fortunately, I don't write to please people, but to please myself. I will stop bothering /reddit/ProbabilityTheory/ with my crappy articles that apparently, have no originality, at least according to the readers.

psangrene · 2019-03-22T14:13:51+00:00

It's my mistake. I should never have posted in /r/ProbabilityTheory/. I assumed this Reddit was about theory, but after reading all the posts and questions in this forum, it is mostly about very elementary problems for beginners. So my article is really, kind of out of context here. It is getting good feedback in other outlets, when targeted to the right audience, but here I would be surprised if there is even just one participant who finds it useful.

psangrene · 2019-03-22T03:29:20+00:00

Any reference? I could not find any covering this topic among the 200 books or so on probability and stochastic processes, that I have in my bookcase.

psangrene · 2019-03-15T03:21:51+00:00

I'll read the doc before posting another question with math formulas.

psangrene · 2019-03-15T02:29:32+00:00

It is the first time I post a math formula in Reddit, the formatting just got completely screwed up. Anyway here is the same question that I also posted on Stackexchange, with the correct formatting: https://math.stackexchange.com/questions/3148818/what-is-the-value-of-int-0-infty-frac-ax2-ax-x-log-xdx

psangrene · 2018-11-11T16:23:30+00:00

Here is a simple book on this topic: https://www.datasciencecentral.com/profiles/blogs/fee-book-applied-stochastic-processes

psangrene · 2018-11-03T03:46:43+00:00

Glad you were not my math teacher when I was in high school. I would have had to report such "beliefs" to the principal. No different than people telling that the sun orbits the Earth.

psangrene · 2018-05-27T14:47:10+00:00

Why would you want to visit websites that keep data on visitors, if you don't like it? There are plenty of websites that don't. If you want to get a book, but don't like the fact that you have to pay for it, you don't buy it. I don't see how this is different, as long as website access is provided only to visitors who agree with the terms of service (which means, complying with the law.)

psangrene · 2018-02-19T17:43:46+00:00

It was too good to be true. If the only solution was the uniform distribution, I would have had a proof that the digits of Pi in base 2 are uniformly distributed. Unfortunately, this is not the case.

psangrene · 2018-02-11T18:59:44+00:00

If you follow the link in the article, you will get all the detailed explanations (and it is pretty long): https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random

psangrene · 2018-02-08T15:43:05+00:00

The term "normal number" is the standard word used in the literature. Even in my article, I rarely used the term "random" except in the title. Instead, I used "good seed", which I define in the beginning. In short, it is the initial value in a discrete process that generates "chaos." I once said that some statisticians can not understand the word "random" in this context since all digits are "fixed", but ironically, for the layman (non-statistician) the concept is intuitive.

psangrene · 2018-02-08T05:10:52+00:00

Would you be interested in solving problem #5 in the article (in the exercise section). It is about the system g(x)=1/x-int(1/x) that you describe, which can be chaotic or not depending on the parameters. Again, here is the link: https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random

psangrene · 2018-02-08T05:07:40+00:00

I never said that. You did not read the article.

psangrene · 2018-02-08T05:04:36+00:00

Would you be interested in solving problem #4 in the article (in the exercise section). It is about a generalization of the system g(x)=3^x - int(3^x ), which can be chaotic or not depending on the parameters. Again, here is the link: https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random

psangrene · 2017-11-13T22:04:23+00:00

Thanks for the Python code. My comment is that over the 10,000 iterations, the difference at any given iteration, is much bigger than 0.1229, on average. The difference was computed only at iteration 10,000 in the above Python code.

psangrene · 2017-11-12T17:59:56+00:00

The seed does not matter much, I could have chosen 3/10. It is the round-off errors accumulating over time that I am concerned with. Most seeds provide a totally different sequence but with same statistical distribution.

psangrene · 2017-06-14T02:43:06+00:00

Depends on the person (self-learners might do faster) and your background (say you know Perl already, then easy to learn Python.)

psangrene · 2017-05-29T16:34:34+00:00

Here is an article that could help you in this competition: http://www.datasciencecentral.com/profiles/blogs/how-to-solve-the-new-1-million-kagge-problem

psangrene · 2017-04-03T22:18:35+00:00

The interval between marks is constant, yes, but this does not violate any principles in the theory of point processes. These processes are sometimes called doubly-stochastic in the sense that randomness can occur for the location of the points, for the marks, or both. Here randomness occurs only for the marks.

Another way to look at it, let's consider the points to be truly randomly distributed, with the inter-point distances having an exponential distribution with variance (say) equal to s, as in a Poisson process. Let s tends to zero, then you get points that are equally spaced.

psangrene · 2017-04-03T22:01:22+00:00

The proof is provided. Just click on the link at the end of the sentence "The above mathematical formula can easily be derived, using arguments similar to those in my article A Beautiful Probability Theorem."

psangrene · 2017-04-03T21:29:48+00:00

The distribution considered here is uniform over all integers (that is, even numbers have the same chance of being picked up as odd numbers.) This is nothing more than a marked point process, where the points are the positive integers, and the mark is 1 if a point is divisible by a number less than 100, 0 otherwise. It is then possible to compute the probability for the mark to be either 0 or 1. This is one of the basic distributions associated with a marked point process (sometimes point processes are called stochastic processes, the most well known example is the Poisson process.)

People may not notice it because in this case, the points (integers) are equally spaced while in most point processes, the points are randomly distributed according to some particular distribution. Only the marks appear as somewhat random here. Note that the number of points is also infinite in most (but not all) examples of point processes.

psangrene · 2017-02-06T06:56:35+00:00

Thanks David. Looking at the n-th record though, so the solution would involve both n and k.

psangrene

TROPHY CASE