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Consider $N = 2^{n}$ random variables $X_{1}, X_{2}, \ldots, X_{N}$, such that for each $i \in [N]$,

$$X_{i} \sim \Gamma\left(\frac{1}{2}, 2^{-n+1}\right). $$

We are also given that $$\sum_{i = 1}^{N}X_{i} = 1$$

What is the pdf for the joint distribution for the $N$-tuple $(X_{1}, X_{2}, \ldots X_{N})$? Does it follow a Dirichlet distribution? Note that the random variables are identically distributed but not independent.

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    $\begingroup$ I can't be: the marginals of a Dirichlet have Beta distributions. $\endgroup$ – Zen Nov 21 '20 at 15:47
  • $\begingroup$ What is the pdf for the joint distribution then? Can it be found out from the given information? $\endgroup$ – BlackHat18 Nov 21 '20 at 15:58
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    $\begingroup$ If the $X_i$'s are iid Gamma's, their sum cannot be one. Do you mean the distribution of the vector conditional to the sum being equal to one? $\endgroup$ – Xi'an Nov 21 '20 at 17:18
  • $\begingroup$ The random variables are identically distributed, but not necessarily independent. So, the sum can be $1$ right? $\endgroup$ – BlackHat18 Nov 21 '20 at 18:19
  • $\begingroup$ I guess I do mean something like "the distribution of the vector conditional to the sum being equal to one." $\endgroup$ – BlackHat18 Nov 21 '20 at 18:20

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