I need a help:

What will be the distribution of sum of $n$ independent F distributed random variables with parameters 1 and 1 (i.e., $F(x;1,1)$?

Formally, say $x_1,\ldots,x_n$ are i.i.d. as F(1,1), what is the distribution of $\sum x_i$?

Great if you can suggest some references too, Thanks in advance.


closed as unclear what you're asking by Denis-Charles Cisinski, Denis Serre, Neil Strickland, Stefan Kohl, Karl Schwede Oct 14 '14 at 17:36

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I doubt that there is a closed form in general. The characteristic function of one of your random variables is, according to Maple, ${{\rm e}^{-is}}{\rm erfc} \left( \sqrt {-is} \right)$, so the characteristic function of your sum is ${{\rm e}^{-ins} }{\rm erfc} \left( \sqrt {-is} \right)^n$. For $n=2$ the PDF for $x > 0$ is $$ \dfrac{2}{\pi (2+x)\sqrt{1+x}} $$ For $n=3$ it is $${\frac {2}{{\pi }^{2} \left( 3+x \right) \sqrt {2+x}} \left( \arctan \left({\frac {{x}^{2}+2\,x-1}{2\;\sqrt {2x+x^2}}} \right) -\arctan \left( {\frac {1}{\sqrt {2x+x^2}}} \right) + \pi \right) } $$

  • $\begingroup$ Thanks a lot. I think it is good enough for me. Just one more question, how do you get the pdf? (e.g., by Maple?) $\endgroup$ – user48855 Oct 15 '14 at 7:30
  • $\begingroup$ Yes, using Maple. $\endgroup$ – Robert Israel Oct 15 '14 at 16:53

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