Sum of two Beta distributed random variables I want to establish (although  I am not certain that it is possible to do so) that, if X,Y  with
$X \sim Beta(\alpha_1, 1- \alpha_1)$
$Y \sim Beta(\alpha_2, 1- \alpha_2)$
then
$X+Y \sim Beta(\alpha_1+\alpha_2, 1-\alpha_1-\alpha_2)$
With the convolution I have:
$
\int_0^1f_Y(z-x)f_X(x)dx
$
$
=\int_0^1 \frac{1}{B(\alpha_1,1-\alpha_1)}\cdot(z-x)^{\alpha_1-1}\cdot(1-z+x)^{-\alpha_1}\frac{1}{B(\alpha_2,1-\alpha_2)}\cdot x^{\alpha_2-1}\cdot(1-x)^{-\alpha_2}
$
$
=\frac{1}{\Gamma(\alpha_1)\Gamma(\alpha_2)\Gamma(1-\alpha_1)\Gamma(1-\alpha_2)}\int_0^1 \cdot(z-x)^{\alpha_1-1}\cdot(1-z+x)^{-\alpha_1}\cdot x^{\alpha_2-1}\cdot(1-x)^{-\alpha_2}
$
I dont see how to continue here.
Any ideas are appreciated! 
Kind Regards 
Humboldt
 A: The flaw in the claim is made apparent by simply considering the support of $X$, $Y$, and the sum $X+Y$:  since $X, Y \in [0,1]$, $X+Y \in [0,2]$ and that is very clearly not Beta distributed no matter what the underlying parameters might be.
A: I have done some simulations of sums of beta rv's, independent but not identically distributed, for some parameter values that seem reasonable for practical estimates of PERT parameters (min, most likely, max, lambda) for project effort estimates.  Since the sum is bounded below by the sum of the mins and similarly for the maxes, it seems reasonable to conjecture that the distribution of the sum is approximately beta.  I find that the approximation is either fairly good or very good for a casual sample of summand distribution parameters.  I fit the approximating beta by doing a least-squares minimization on the PERT most-likely and lambda parameters with Excel Solver.  This technique is of practical value in doing Monte Carlo simulations of project cost.  The cost of a phase is the sum of a number of PERTs for the tasks, and the cost of the project is the sum of the PERT (beta) approximations for the phases.
