# Joint distribution given normalized gamma distributed components

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.

• 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
• 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