TITLE: Sum of n i.i.d Beta-distributed variables
QUESTION [9 upvotes]: Assume that I have $n$ variables that are each $X_i \sim \text{Beta}(\alpha, 1)$ distributed (with the same $\alpha$, i.i.d.).
Is there anything known about the distribution of the sum $Y=\sum_i X_i$?
I've tried to understand the works of Pham-Gia, but couldn't entirely grasp them. He seems to use some modified generalized Beta $B^*$.
An approximation is also fine with me, as long as I can compute some quantiles. I'm mostly interested in quantiles.

REPLY [9 votes]: Your $Y$ will have mean $\frac{n\alpha}{\alpha+1}$ and variance $\frac{n\alpha}{(\alpha+1)^2(\alpha+2)}$
If you want a reasonable approximation other than a normal distribution with that mean and variance, you could say that $\frac{1}{n}Y$ has mean $\frac{\alpha}{\alpha+1}$ and variance $\frac{\alpha}{n(\alpha+1)^2(\alpha+2)}$ and then find a Beta distribution with those moments that would give you $\alpha'=\alpha\frac{\alpha n+2n-1}{\alpha+1}$ and $\beta'=\frac{\alpha n+2n-1}{\alpha+1}$.  
The density function for this approximation to $\frac{1}{n}Y$ would then be $\frac{\Gamma(\alpha'+\beta')}{\Gamma(\alpha')\Gamma(\beta')}x^{\alpha'-1}(1-x)^{\beta'-1}$ on $[0,1]$ and the density function for the corresponding approximation to $Y$ would be $\frac{\Gamma(\alpha'+\beta')}{n\Gamma(\alpha')\Gamma(\beta')}\left(\frac{x}{n}\right)^{\alpha'-1}\left(1-\frac{x}{n}\right)^{\beta'-1}$ on $[0,n]$. 
As an example, consider $\alpha=3$ and $n=2$.  You get a mean of $\frac32$ and variance of $\frac3{40}$, and for the approximation you get $\alpha'= \frac{27}{4},\beta'= \frac{9}{4}$.  In the following plot, the black curve gives the actual density, the red curve the stretched-Beta approximation and the blue curve the normal approximation.  The red curve is not too far away in this case. 

Similarly the cumulative distributions look like the next chart and you can see that the quantiles of the red stretched-Beta approximation are much closer than the quantiles of the blue normal approximation here.