Math 6010 • Linear Models

Fix \(0<\lambda <\rho \) and let \(X\), \(Y\), and \(Z\) be three independent Gamma distributed random variables with common scale parameter 1 and shape parameters \(\lambda \), \(\rho -\lambda \), and \(\rho \), respectively. (A Gamma random variable with scale parameter 1 and shape parameter \(\lambda \) has pdf \(\frac {1}{\Gamma (\lambda )}x^{\lambda -1}e^{-x}\mathbf {1}_{\{x>0\}}\).) Let \[ U=\frac {ZX}{X+Y},\qquad V=\frac {ZY}{X+Y},\qquad \text {and}\qquad W=X+Y. \] Prove that random vector \((U,V,W)\) has the same distribution as \((X,Y,Z)\). (It may be useful to note that \(Z=U+V\).)