I need to prove X = (X1, X2, X3) is independent where

X1 = Z1 / (Z1+Z2)
X2 = (Z1+Z2) / (Z1+Z2+Z3)
X3 = (Z1+Z2+Z3) / (Z1+Z2+Z3+Z4)

and Z ~ G(1)

What I did was, first find X1 ~ beta (4,2), X2 ~ beta(5.3), X3 ~ beta(6,4) by using following theorem:

Z~G(p), W~G(q), Z and W are independent
--> Z+W ~ G(p+q)
--> (Z+W) and u are independent where u=Z/(Z+W)

However, how can i prove X=(X1, X2, X3) is independent with these information????? or Am I going a wrong direction????

Please help me!!!