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Suppose I have 3 random variables, $X1, X2,X3$. Define $Z$ as:

$Z=X1+X2+X3$

I want to force $Z$ to equal 1 for every "realization" of $X1,X2,X3$ ($X_i \sim Beta(a_i,b_i))$. As an example, let $X_i$ denote the percentage of votes a party will get. In a hypothetical country, there are only 3 parties. Therefore, $Z$ is the total percentage of votes, which equals to 100%. Statistically, how could someone force Z to equal 1, and by simulation or through an analytical expression derive the percentages of party A,B and C?

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  • $\begingroup$ I am not sure if that's enough, but can you make just C=1-A-B? $\endgroup$ – David May 29 '19 at 7:11
  • $\begingroup$ Thought about your answer, but there might be realization, where C would turn to negative (e.g if parties A and B get most of the votes). Also, C might not keep its distributional properties. (I try to think of it as a transformation) $\endgroup$ – AlexandrosB May 29 '19 at 7:16
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    $\begingroup$ Your normals could be outside the intervals $[0,1]$, which makes the interpretation as fractions of votes hard. If that is a problem, you may want to ask about beta distributions, or look for random generation on a simplex. $\endgroup$ – Stephan Kolassa May 29 '19 at 7:18
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    $\begingroup$ @David There are two problems with your suggestion. The first is that $C$ could be negative. This reveals the second problem: $C$ will not have a Beta distribution. This is a subtle issue, to be sure: we get many questions of this nature that ask for a set of variables to have given marginal distributions subject to one or more linear relations among them. Most often such variables don't even exist. $\endgroup$ – whuber May 29 '19 at 11:43
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    $\begingroup$ @David Sorry about that--I have often worried about how edits to questions can leave comments (and even answers) that make sense but don't have their original meaning. In order to avoid this problem I have been making efforts to quote or restate assertions I am responding to, if there's room and I remember to do it! $\endgroup$ – whuber May 29 '19 at 11:56
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What you want is the Dirichlet distribution.

It is the generalisation of the Beta distribution to the multivariate case, and provides samples under the constraint that each $X_i \in (0, 1)$ and $\sum_{i=1}^3 X_i = 1$, always. In addition, you also get interpretable Beta marginals from the distribution.

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  • $\begingroup$ I have estimated the scale parameters of the beta distribution for each random variable $X_i$. How could I link these parameters with the concentration parameters of the Dirichlet distribution? $\endgroup$ – AlexandrosB May 29 '19 at 18:05

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