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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.

BlackHat18
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    I can't be: the marginals of a Dirichlet have Beta distributions. – Zen Nov 21 '20 at 15:47
  • What is the pdf for the joint distribution then? Can it be found out from the given information? – BlackHat18 Nov 21 '20 at 15:58
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    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? – Xi'an Nov 21 '20 at 17:18
  • The random variables are identically distributed, but not necessarily independent. So, the sum can be $1$ right? – BlackHat18 Nov 21 '20 at 18:19
  • I guess I do mean something like "the distribution of the vector conditional to the sum being equal to one." – BlackHat18 Nov 21 '20 at 18:20
  • The joint distribution does not have a PDF because it is singular. Perhaps you would like the PDF of $(X_1, X_2, \ldots, X_{N-1})$ instead? For a standard relationship between Gamma variates and the Dirichlet distribution, please see https://stats.stackexchange.com/questions/36093. And for a very closely related question, see https://stats.stackexchange.com/questions/252692. (Notice that when you take sums of your $X_i$ in disjoint pairs you have $N/2$ exponential variables, exactly as posited there.) – whuber Nov 21 '20 at 20:26
  • A few confusions: why is the joint distribution singular? In the second link, you seemed to imply that the $N$-variate distribution is a symmetric Dirichlet. If that is so, does that not mean that the $N$-variate distribution has a PDF? Also, in the second link, the $X_{i}$s seemed to have been independent. Will that be a problem for my case? Finally, what is the PDF for the $(N-1)$-variate distribution, like you said? Will it also be a Dirichlet (I can't immediately relate the same proof techniques as the random variables are no longer independent.) – BlackHat18 Nov 22 '20 at 10:07
  • Also, in the second link you provided each $X_{i}$ had an exponential distribution, whereas for me, I have a Gamma distribution. How will that change the final result? – BlackHat18 Nov 22 '20 at 15:55
  • See https://stats.stackexchange.com/questions/36093/construction-of-dirichlet-distribution-with-gamma-distribution/154298#154298. – whuber Nov 23 '20 at 14:46

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