# Distribution of $\frac{X}{X+Y}$ if $X,Y$ are independent Beta random variables

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?

• It is not a Beta distribution: experiment with $a_1=3,a_2=1$ and you will see something unlike any Beta distribution – Henry May 18 at 12:01
• It has a unique mode in $(0,1)$ and non-zero density at 0? – econ86 May 18 at 12:23
• 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. – whuber May 18 at 13:51
• 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 – Henry May 18 at 14:13
• Special case: math.stackexchange.com/questions/541322/… – Peter O. May 18 at 17:46