Interesting Esoterica

On Buffon Machines and Numbers

Article by Flajolet, Philippe
  • Published in 2011
  • Added on
The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.

Links

Other information

journal
Bernoulli
pages
1--12

BibTeX entry

@article{Flajolet2011,
	title = {On Buffon Machines and Numbers},
	author = {Flajolet, Philippe},
	url = {http://arxiv.org/abs/0906.5560 http://arxiv.org/pdf/0906.5560v2},
	journal = {Bernoulli},
	pages = {1--12},
	year = 2011,
	urldate = {2011-01-12},
	abstract = {The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi {\~{}} 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.},
	archivePrefix = {arXiv},
	eprint = {0906.5560},
	collections = {Easily explained,Probability and statistics}
}