We know the well-known beta distribution and gamma distribution. I am just wondering why don't we have alpha distribution? I am guessing there must be a reason for this but couldn't find it elsewhere.
*For beta and gamma distributions, I am guessing that since beta distribution involves beta function and gamma distribution involves gamma function, so their distributions are named after these special functions.
Any comments/answers will be very appreciated.
On
These distributions are probably named after the special functions, which leads to the question: "was there an alpha function"? This has been asked before on this site. Also, in general there seems to be an aversion to using vowels as names of mathematical objects.
There is no such thing as an alpha function, and there is no such thing as an alpha distribution. That we have beta and gamma distributions is mere coincidence, just like how Gamow was too playful and inserted Bethe's name into a paper he only co-authored with Alpher.
However, there are the Lévy alpha-stable distributions on four parameters, one of which is $\alpha$, which controls the most important stability parameter of the distribution.