Given $\textbf{P}$ independent and identically distributed random variables, $X\_1, X\_2, ..., X\_P \sim \Gamma(M,2c)$ how can we prove that:


$$U = X\_1 + X\_2 + ... + X\_P$$ 


and 


$$V = \frac{X\_1}{X\_1 + X\_2 + ... + X\_P}$$


are independent?


Where $U \sim \Gamma(MP,2c)$ and $V \sim \beta(M,M(P-1))$.



Independence of Gamma and Beta random variables with common term
Just for the record: this is answered in item 25 of <http://www.randomservices.org/random/special/Beta.html> 


