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If we have a vector of probabilities $\textbf{p} = (p_1,...,p_n)$, with $p_i \in [0,1]$ and $\sum p_i = 1$, is it possible to model $\textbf{p}$ as a joint random variable with some multivariate distribution?

Each $p_i$ itself could perhaps be modeled in a variety of ways (although the lower and upper bounds complicate things a bit), but how could we possibly deal with the fact that they must sum to $1$ if we want a joint distribution?

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    The multivariate distribution that you are supposed to look at is the Multinomial distribution. Look in https://en.wikipedia.org/wiki/Multinomial_distribution, this preserves the sum to be $1$. – 2017-02-11
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    It is possible, it is commonly done, a frequently used choice to do this is called [Dirichlet distributions](https://en.wikipedia.org/wiki/Dirichlet_distribution). – 2017-02-11

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