Zipf's law, Benford's law, and more
Suppose you’re listening to radio station ZIPF, a station that randomly plays the top 100 songs according to Zipf’s law: the nth most popular song is played with probability proportional to 1/n. How long would it take on average to hear all 100 songs at least once?
Zipf’s law comes up often. A lot of things, such as word frequencies, approximately follow Zipf’s law over a wide part of their range.
The first digits of numbers are not usually uniformly distributed. This was first discovered when someone noticed that the pages in a book of logarithms were dirty at the beginning and clean at the end. People more often looked up the logs of numbers starting with 1 than numbers starting with 9.
Benford’s law does a good job of estimating the distribution of first digits in certain sequences of numbers. Sometimes it’s even exact.
A few days ago, the exponential sum of the day had an unusual number of horizontal and vertical lines. I explain why here.
The latest post looks at a logarithmic sawtooth, a function whose plot on a log scale is an infinite sawtooth wave. Surprisingly the function can be integrated in closed form, though it’s requires some finesse to integrate numerically.
Thanks for reading. Have a good weekend.