# Independence of some derived random variables

Positive random variables $X_1, X_2,$ and $X_3$ have joint probability density give by $$f_{\lower{0.5ex}{X_1,X_2,X_3}}\!(x_1,x_2,x_3)= \begin{cases}48~x_1~x_2~x_3 & : \textsf{if }0<x_1<x_2<x_3<1\\[1ex]0 & :\textsf{otherwise.}\end{cases}$$

Are the new variables $Y_1=X_1/X_2, Y_2=X_2/X_3 \text{, and } Y_3=X_3$ statistically independent?

By observing that the transformation between $X_i$ and $Y_i$ is reversible, you can calculate the density $f'$ on a triplet of $Y_i$ by dividing the value of f on the corresponding triplet on $X_i$ by the determinant of the Jacobian matrix $J= \frac{\partial X}{\partial Y}$

It is easy to show that :

$J= \begin{bmatrix} Y_3 Y_2 & . & . \\ 0 & Y_3 & . \\ 0 & 0 & 1 \end{bmatrix}$

so $\det(J)=Y_3^2 Y_2$

and $f'_{Y_1,Y_2,Y_3}=f_{Y_1Y_2Y_3,Y_1Y_2,Y_3}*\det(J)= 48 * Y_1Y_2Y_3$

The domain of definition of Y is simplier than that of X as it is the cartesian product $]0,1[ \times ]0,1[ \times ]0,1[$

So the density function is a product of functions over each $Y_i$ and so those variables are independent.