Cauchy, Benford, and a problem with NHST

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Technical notes on math and software

Table of contents Math diagrams Numerical computing Probability Differential equations Python Probability approximations Regular expressions C++ Special functions Typesetting: TeX, HTML, Unicode Emacs R Miscellaneous math My notes on cryptography have their own page. Math diagrams Diagram of probability distribution relationships Modes of convergence Topological properties diagram Category Relationships in Mathematical Physics Category theory definition […]

Leading digits and quadmath

My previous post looked at a problem that requires repeatedly finding the first digit of kn where k is a single digit but n may be on the order of millions or billions. The most direct approach would be to first compute kn as a very large integer, then find it’s first digit. That approach […]

Gelfand’s question

Gelfands’s question asks whether there is a positive integer n such that the first digits of jn base 10 are all the same for j = 2, 3, 4, …, 9. (Thanks to @republicofmath for pointing out this problem.) This post will explore Gelfand’s question via probability. The MathWorld article on Gelfand’s question says that […]