Tuesday, April 10, 2018

Two Beautiful Mathematical Results - Part 2

In Part 1 of this article (see here) we featured the two results below, as well as a simple way to prove these formulas.
Here, we continue on the same topic, featuring and proving the formulas below, which are just the tip of the iceberg.
However cool these formulas might look, the biggest contribution here is a general framework to solve much more general problems of this type. The mathematical level is still relatively simple, accessible to people in their first year of college education, if they attended a solid course on calculus. These results are still easy to prove for people who have been exposed to the basics of complex number theory.
I haven't seen these results published anywhere, but my guess is that they are not new. I encourage readers to post questions on Quora or Stackexchange to find more references on this topic, as Google search is of no use here.The focus here is to get more people interested in mathematics, by featuring fascinating results that are not that hard to prove, even for high school students participating in mathematical Olympiads. Also, it could be an interesting, fresh, original topic for university professors to discuss in their lectures or for exams. Finally, if you have seen many similar formulas on Wikipedia (or elsewhere), and you are wondering how they are derived, you will find the solution in this article.
The last section introduces a new, tough, unrelated problem, still unsolved today, that will be of interest to people with a background in probability and/or number theory.

Sunday, April 1, 2018

I Analyzed 10 MM digits of SQRT(2) - Look at My Findings

This article is intended for practitioners who might not necessarily be statisticians or statistically-savvy. The mathematical level is kept as simple as possible, yet I present an original, simple approach to test for randomness, with an interesting application to illustrate the methodology. This material is not something usually discussed in textbooks or classrooms (even for statistical students), offering a fresh perspective, and out-of-the-box tools that are useful in many contexts, as an addition or alternative to traditional tests that are widely used. This article is written as a tutorial, but it also features an interesting research result in the last section. The example used in this tutorial shows how intuiting can be wrong, and why you need data science.
The main question that we want to answer is: Are some events occurring randomly, or is there a mechanism making the events not occurring randomly? What is the gap distribution between two successive events of the same type? In a time-continuous setting (Poisson process) the distribution in question is modeled by the exponential distribution. In the discrete case investigated here, the discrete Poisson process turns out to be a Markov chain, and we are dealing with geometric, rather than exponential distributions. Let us illustrate this with an example.
Example
The digits of the square root of two (SQRT(2)), are believed to be distributed as if they were occurring randomly. Each of the 10 digits 0, 1, ... , 9 appears with a frequency of 10% based on observations, and at any position in the decimal expansion of SQRT(2), on average the next digit does not seem to depend on the value of the previous digit (in short, its value is unpredictable.)  An event in this context is defined, for example, as a digit being equal to (say) 3. The next event is the first time when we find a subsequent digit also equal to 3. The gap (or time elapsed) between two occurrences of the same digit is the main metric that we are interested in, and it is denoted as G. If the digits were distributed just like random numbers, the distribution of the gap G between two occurrences of the same digit, would be geometric
Do you see any pattern in the digits below? Read full article here to find the answer, and to learn more about a powerful statistical technique.

Two Beautiful Mathematical Results - Part 2

In Part 1 of this article (see   here ) we featured the two results below, as well as a simple way to prove these formulas. Here, we co...