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.
 A: 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.
