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$X_1,X_2,X_3$ 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.

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up vote 2 down vote accepted

As soon as you see that the joint density factors as $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3) = g(y_1,y_2) h(y_3)$ for some functions $g$ and $h$, it's clear that $(Y_1,Y_2)$ and $Y_3$ will be indepedent: integrate with $dy_1\ dy_2$ or $dy_3$ respectively to get that $f_{Y_1,Y_2}(y_1,y_2)$ and $f_{Y_3}(y_3)$ are constant multiples of $g(y_1,y_2)$ and $h(y_3)$, and then $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3) = f_{Y_1,Y_2}(y_1,y_2) f_{Y_3}(y_3)$.

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