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Let $X \sim \text{Beta}(a,b)$ and $Y \sim \text{Beta}(b,a)$ be independent random variables.

What is the distribution of $\frac{X}{X+Y}$? Could it be Beta itself?

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    $\begingroup$ It is not a Beta distribution: experiment with $a_1=3,a_2=1$ and you will see something unlike any Beta distribution $\endgroup$ – Henry May 18 at 12:01
  • $\begingroup$ It has a unique mode in $(0,1)$ and non-zero density at 0? $\endgroup$ – econ86 May 18 at 12:23
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    $\begingroup$ It can have as many as three modes. Try, e.g., $a=b=1/2.$ It can have zero density at $0$ and $1:$ try $a\ge 2$ and $b\ge 2.$ There's no "nice" closed formula for the density function. $\endgroup$ – whuber May 18 at 13:51
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    $\begingroup$ I meant that experimenting showed a density shape which is clearly not a Beta distribution. @whuber 's example is even more obviously not a Beta distribution $\endgroup$ – Henry May 18 at 14:13
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    $\begingroup$ Special case: math.stackexchange.com/questions/541322/… $\endgroup$ – Peter O. May 18 at 17:46

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