The Gamma function for positive arguments can be defined with the integral

$$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx $$

The function $ x^{\alpha-1} e^{-x} $ is called the Gamma distribution when normalized.

How are the two names related?

  • Did the Gamma function get this name and the $\Gamma$ symbol because the distribution was already called Gamma distribution?
  • Or backwards, did the Gamma distribution get named from the Gamma function?
  • Or were the two named together at the same time?
  • Or are the two names independent and only accidentally the same letter, despite the apparent connection?

This question came up in a discussion about spelling mathematical terms, and while it's probably not actually relevant for that, I'm now curious about the origin of the names.


According to Herbert A. David’s First (?) Occurrence of Common Terms in Statistics and Probability (2010 revision), “Gamma distribution” goes back to Paul G. Hoel, A significance test for component analysis, Annals of Mathematical Statistics 8 (1937) 149–158. That’s much later than Legendre’s naming of the Gamma function (see Wikipedia).


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