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Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf $Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma (\alpha_i)}\beta_i^{-\alpha_i} x^{\alpha_i -1} e^{-x/\beta_i}$, with shape-parameters $\alpha_i$ and scale-parameters $\beta_i$.

Now if $\beta_i=1$ for $i=1..N$, it is known that the transformed multivariate random variable $\{Y_i\}_{i=1..N}$ defined by $Y_i := \frac{X_i}{\sum_{i=1}^N X_i}$ is distributed according to an N-dimensional Dirichlet-distribution with shape parameters $\{\alpha_i\}_{i=1..N}$.

I was wondering if a similar result (meaning closed form expression) for the distribution of the $\{Y_i\}_{i=1..N}$ can be obtained by fixing the shape-parameters to $\alpha_i = 1$ and allowing arbitrary scale-parameters $\beta_i$ ?

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