Approximate the distribution of the sum of ind. Beta r.v If $X_i$ has a Beta distribution $\beta(1,K)$.
What is the best approximation for the distribution of $ S=\sum_{i=1}^N X_i$, when the $X_{i}$ are independent and $N$ is finite.
 A: Since you cannot share details, we cannot know to what extend you can use transformations of your variables, and still get what you need. For what is worth, a usual transformation here is the following:
$$X_i \sim \beta(1,K) \Rightarrow Z_i=-\ln(1-X_i) \sim \text{Exp}(K)$$
and
$$S_z = \sum_{i=1}^NZ_i \sim \text{Erlang}(N,K)$$
with pdf
$$f_{S_z}(s_z) = \frac {K^NS_z^{N-1}e^{-KS_z}}{(N-1)!}$$
A: CLT: $(S_n-n\mu)/\sigma\sqrt{n}$ is approximately distribured as $\mathrm{N}(0,1)$ for large $n$.
To determine $\mu$ and $\sigma$ use this and this.
A: If you want better approximations than what you get from the central limit theorem, there is results in a book dedicated exclusively to the  beta distribution:   http://www.amazon.com/Handbook-Beta-Distribution-Applications-Statistics/dp/0824753968/ref=sr_1_1?s=books&ie=UTF8&qid=1403444915&sr=1-1&keywords=beta+distribution
(On the amazon.com website you can search within this book!) Arouind page 70 there is exact results for the sum of two independent beta distributions, arouind page 70 they find an approximation by assuming the sum also has an generalized beta distribution, and then equate moments. On page 85 they give approximations for general sums by using the same method, equating moments. Around page 85-87 they give references you can follow up.
