On wikipedia (https://en.wikipedia.org/wiki/F-distribution) I see the pdf of F distribution defined using squared root function, while other places I see the pdf defined with the gamme function.

E.g from the book Modern Mathematical Statistics with Applications :

"The pdf of a random variable having an F distribution is:"

$g(x) = \frac{{\Gamma \left( (v_1 + v_2)*0.5 \right)}} {{\Gamma \left( (v_1)*0.5 \right) +\Gamma \left( (v_2)*0.5 \right)}} (\frac{{v_1}}{{v_2}})^{v_1/2} * \frac{{X^{(v_1/2) -1} }}{{(1+v_1x/v_2)^{(v_1+v_2)/2}}} $

  1. How does one derive the pdf of the F distribution which uses the gamme function as defined above to a pdf that uses the Beta function as defined in the wikipage for the F distribution?

There is also the definition given two chi-squared random variables $X_1$ and $X_2$ that defines:$F_{m-1, n-1}$ but sometimes there is a different definition $F_{p, m-1, n-1}$.

  1. What is the difference between these two definitions and how are they related to the pdf?

Edit: The question has been closed with “unclear what you are asking”. So I have clearified the 2 questions. They are maybe trivial to some, but not to me as a beginner.

  • $\begingroup$ Could you show us one of the definitions in terms of Gamma functions so that we can understand what you are asking? $\endgroup$ – whuber Feb 4 at 15:36
  • $\begingroup$ That is g(x) as defined above given by the book Modern Mathematical Statistics. The other definition is on wikipedia as linked to above. $\endgroup$ – Endre Moen Feb 5 at 16:18
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    $\begingroup$ The definitions are identical: the Wikipedia article uses the Beta function as a shorthand for the ratio of gamma functions appearing in your definition (which, btw, is misquoted--but that doesn't matter; the idea is clear). $\endgroup$ – whuber Feb 5 at 16:59
  • $\begingroup$ Ok - so that is a new idea to me. Didnt know the Beta function was shorthand for the gammefunction. What about the square root in the numerator of the wiki definition? There is no square root in the numerator of the gamma-function-definition. The equivelance isnt obvious to me. $\endgroup$ – Endre Moen Feb 6 at 11:58
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    $\begingroup$ Sure: see the Wikipedia article on the Beta function, for instance. $\endgroup$ – whuber Feb 6 at 15:40