Addition of distributions in statistics

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Is it possible to add distributions? I've worked out "Say that you are given ten identical coins for which you assume Beta(4,4) prior distribution on the unknown probability θ of any of the coins showing head. Someone tosses these ten coins and tells you that fewer than three heads showed (but you do not know whether it is zero, one, or two heads)", as Beta(6,12)+Beta(5,13)+Beta(4,15). I'm not certain how to draw all of them together to form one distribution, I think it might be using the Central Limit Theorem but I'm not sure how to apply it

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Yes you can but you need normalization such that the sum integrates to one. For example, if you have $n$ distributions denoted as $X_1, X_2, \dots, X_n$, then

$$ Y=a_1X_1+a_2X_2+\cdots+a_nX_n $$

is a proper distribution if $\sum_{i=1}^n a_i=1$. Then you can sample from the $X_i$ distributions and multiply them by their weights.