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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 $\sum_i \omega_i = 1$?

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