# Sum of Beta Random Variables

Supose I have $$I$$ independently (but not necessarily identically) distributed beta random variables, $$X_i = \text{beta}(\alpha_i,\beta_i)$$, for $$i=1,\dots,I$$.

Is there a known distribution for the sum of these r.v.s, $$Y=\sum_i X_i$$?

Furthermore, is there a known distribution for the mixture of these r.v.s, as in $$Y=\sum_i \omega_i X_i$$, where $$\sum_i \omega_i = 1$$?

• – Gabriel Romon Feb 1 '19 at 16:08
• Not in closed form. There isn't even a closed form expression for the PDF of the sum in the case $n=2$. – Robert Israel Feb 1 '19 at 16:26