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November 11th, 2017, 04:45 PM   #1
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Exclamation Function of Two Independent Gammas follow Beta



I've gotten so close but I cannot figure out for the life of me what's wrong.

At the end, my PDF comes out to be:



I'm literally off by the constant of theta. For my substitutions, I let U = X + Y and W = X/(X+Y) and my Jacobian came out to be u (after taking the absolute value). So then I just do the substitutions and multiply by the Jacobian, and I get what I have above... which is NOT the Beta Distribution PDF since I'm off by the constant of theta. Can somebody show me a proof of this so I can see where I'm going wrong?
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November 12th, 2017, 12:26 AM   #2
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https://www.math.ucdavis.edu/~romik/...iscussion3.pdf

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November 12th, 2017, 08:37 AM   #3
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Thanks, I was just failing to do my integral substitutions properly. That's why my theta wasn't disappearing.
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