# Why is the Beta Distribution Called the Beta Distribution?

In my research, I've been unable to find a historical explanation for why the Beta distribution has the name it does. I'm aware of what Wikipedia says about how it got its name, but so far I haven't found anything that would make its name appear to be anything but arbitrary. (Why beta and not lambda or upsilon or the Boise, Idaho distribution ;-) ?) I'm aware of the history section in the Wikipedia article; it addresses the how but not the why.

• Hint: Look at the probability density function. – soakley Apr 28 '20 at 17:56
• This might be a chicken/egg scenario, but see this: en.wikipedia.org/wiki/Beta_function – Demetri Pananos Apr 28 '20 at 18:03
• Wikipedia claims it was named by Jacques Binet, but provides no reference. Certainly its study and naming by mathematicians preceded its use by statisticians. I would venture to guess that the name has nothing at all to do with the shape of the PDF and probably everything to do with the naming of several related functions at once, of which the Beta and Gamma persist. – whuber Apr 28 '20 at 18:05
• jeff560.tripod.com/functions.html supports Beta from Binet. – Nick Cox Apr 28 '20 at 18:16
• Facetious history beats factual history, but I think the credit for gamma function belongs to Euler and Legendre. Euler used it and Legendre named it, as I recall. The gamma distribution comes later. – Nick Cox Apr 28 '20 at 18:44

Florian Cajori, in History of Mathematical Notations Vol. II (1928), wrote

... in the same paper of 1730 Euler gave what we now call the "beta function." ... About a century after Euler's first introduction of this function, Binet wrote the integral in the form $$\int_0^1 x^{p-1} dx(1-x)^{q-1}$$ and introduced the Greek letter beta, $$B.$$ Considering both beta and gamma functions, Binet said: "Je désignerai la première de ces fonctions par $$B(p,q);$$ et pour la seconde j'adopterai la notation $$\Gamma(p)$$ proposée par M. Legendre." Legendre had represented the beta function by the sign $$\left(\frac{p}{q}\right).$$

(Translation: I will call the first of these functions $$B(p,q);$$ and for the second I will adopt the notation $$\Gamma(p)$$ proposed by Mr. Legendre.)

Cajori references Jacques P. M. Binet in Journal de l'Ecole Polytechnique, Vol. XVI (1839), p. 131.

A Web page maintained by St. Andrews (Scotland) School of Mathematics and Statistics relates that Binet

wrote Mémoire sur les intégrales définies eulériennes et sur leur application à la théorie des suites; ainsi qu'à l'évaluation des fonctions des grands nombres in 1839. In this paper Binet introduced what today is called the Beta function $$B(m,n).$$ It has been suggested that Binet chose the notation $$B$$ and called it a beta function, because of the first letter of his own name. However, there is no evidence to support this claim.

(If I may speculate, I would propose that having placed the two functions in order, Binet selected $$B$$ as the antecedent letter in the Greek alphabet to $$\Gamma$$--and might not have minded that it was also his initial.)

A promising reference I came across is a history of the Gamma function: M. Godefroy, La fonction Gamma; Théorie, Histoire, Bibliographie, Gauthier-Villars, Paris (1901), but I haven't searched out a copy.

In his History of Statistics (1986), Stephen Stigler relates that Thomas Bayes worked with the Beta function:

The evaluation of the integral $$\int_0^f \theta^p(1-\theta)^q\,d\theta,$$ Bayes noted, ... would complete the solution. This integral is now known as the incomplete beta function ... . The first extensive tables of this function were not compiled until this century [20th], when the students in Karl Pearson's laboratory were pressed into reluctant service as "computers." A story, possibly apocryphal, still circulates in University College London of a student who resigned in disgust after a week, telling Pearson of his plans for a different career and announcing, "As far as I am concerned, the Table of the Incomplete Beta Function may stay incomplete."

[at p. 130]. Stigler dates this to c. 1755 [at p. 123], placing it a generation after Euler's paper (q.v.) but almost a century before Binet named it. It doesn't appear that Bayes gave this function any special name, but it is interesting that it had already emerged as important in statistical investigations.

• Thanks; that was very interesting. If I may try to summarize: although various explanations for the choice of name for the beta distribution have been offered, they are speculative. We simply don't have have enough evidence to know, with confidence, why Binet chose to name the distribution the beta distribution. – Jeff Lowder Apr 28 '20 at 20:30
• I agree, Jeff: but at least we know who and when and have some historical context. As a bonus, we have the anecdote at the end of the Stigler quotation :-). – whuber Apr 28 '20 at 22:50
• Thank you, @Xi'an, for supplying the accents--I don't have them on my keyboard and hadn't gotten around to inserting them all. I plead laziness, not ignorance! – whuber Apr 28 '20 at 22:54
• @Patrick Yes, I'm sure it was. I just couldn't think of a simple, clear way to indicate that I believed the error was Cajori's and not Binet's, nor did I (or do I yet) have access to Binet's paper to confirm this belief. So I let the "sic" stand as an accurate representation of my source. – whuber Apr 29 '20 at 12:10
• @Patrick Wonderful! Thank you for finding it. – whuber Apr 29 '20 at 23:06