$X_1,X_2,X_3$ be independent variables, and have marginal pdfs $f_{x_i}(x_i)=c_ix_i^ie^{-x_i}$, $x_i>0$. Let $Y_1=X_1/(X_1+X_2+X_3),Y_2=X_2/(X_1+X_2+X_3),Y_3=X_1+X_2+X_3$. How can I know if $(Y_1,Y_2)$ and $Y_3$ are independent?
I have worked out the $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3)=c_1c_2c_3y_1y_2^2(1-y_1-y_2)^3y_3^8e^{-y_3 }$. But in the solution to this question, it gets to $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3)=f_{Y_1,Y_2}(y_1,y_2)f_{Y_3}(y_3)$ immediately. Could someone explain this to me. Thanks in advance.