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I'm wondering if there's a general formula to simplify the mixture of two beta distributions.

Ex. I have $.5\operatorname{beta}(a_1,b_1) + .5\operatorname{beta}(a_2,b_2)$. Can I find $a_3$ and $b_3$ such that this mix is equivalent to a $\operatorname{beta}(a_3,b_3)$?

Thanks

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Suppose $a_1=1000$ and $b_1=1$, and $a_2=1$ and $b_2=1000$.

Then the average of the two distributions is bimodal with maximum density near but not at $0$ and $1$. Beta distributions are not bimodal except when the parameter values are less than $1$ and the two modes are exactly $0$ and $1$.

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No, the mixture of two beta distributions is not a beta distribution.

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