Suppose $S_x\sim\Gamma(n,\theta)$,$S_y\sim\Gamma(m,\theta)$, $S_x$ and $S_y$ are independent. What's the distribution of $T=\frac{S_x}{S_x+S_y}$?
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2Hint: You can find an answer at https://stats.stackexchange.com/a/200639/11887 – kjetil b halvorsen Jul 20 '21 at 23:12
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3You can also simply look at the "related distributions" section of the wikipedia page on the gamma distribution for the same answer. ... Is this coursework? – Glen_b Jul 20 '21 at 23:54
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1https://stats.stackexchange.com/questions/88628 is a special case. When $n$ and $m$ are integral, the method described at https://stats.stackexchange.com/questions/252692 gives a solution. – whuber Jul 21 '21 at 12:43
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@kjetilbhalvorsen Thanks! That's very helpful. – ExcitedSnail Jul 22 '21 at 02:46
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@Glen_b Thanks! It's very helpful! – ExcitedSnail Jul 22 '21 at 02:46
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@whuber Thanks! Yes, m and n are integers. – ExcitedSnail Jul 22 '21 at 02:46