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Suppose we have $X_{1} \sim B(m,p_{1}), X_{2} \sim B(m,p_{2}),\cdots, X_{n} \sim B(m,p_{n})$ and they are dependent. Does the joint distribution $f(X_{1},X_{2},\cdots,X_{n}) $ have a closed form?

Edit: let's take as an example a random graph, what's the joint distribution of the degrees of an ER random graph?

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    $\begingroup$ Something more than 'they are dependent' needs to be assumed for the joint distribution to even be uniquely defined. $\endgroup$
    – Juho Kokkala
    Aug 25, 2016 at 14:15
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    $\begingroup$ If that's what you are interested in, could you perhaps make the question to be about whether the joint distribution of the degrees of an Erdős–Rényi (I presume) graph is available in closed form? (I don't think there can be any general answer). $\endgroup$
    – Juho Kokkala
    Aug 25, 2016 at 14:22
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    $\begingroup$ Or perhaps it's better to post a new question since this already has an answer that's getting upvotes. $\endgroup$
    – Juho Kokkala
    Aug 25, 2016 at 14:28

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There is no unique joint distribution. In fact, there are infinite possibilities to construct the joint distribution. For instance, there exist infinitely many copula functions that can be used to construct a joint distributions with such marginals.

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    $\begingroup$ +1 But note that many of those copulae produce the same joint distribution. It might be clearer to appeal to the discreteness of this distribution: when $n\ge 2$, it is determined by $(m+1)^n$ numbers (all in the interval $[0,1]$) subject to $n m + 1$ independent linear constraints (to match the marginal distributions and to sum to unity). That means $f$ has $(m+1)^n-(nm+1)$ free parameters. $\endgroup$
    – whuber ♦
    Aug 25, 2016 at 14:23

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