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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.

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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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