91Ó°ÊÓ

Define the transformation of variables as follows:\n\[\begin{{align*}}Y_1 &= X_1 / X_2 ,\Y_2 &= X_2 / X_3 ,\Y_3 &= X_3 / X_4 ,\Y_4 &= X_4 .\\end{{align*}}\]

Find the inverse transformation. These are:\n\[\begin{{align*}}X_1 &= Y_1 Y_2 Y_3 Y_4,\X_2 &= Y_2 Y_3 Y_4,\X_3 &= Y_3 Y_4,\X_4 &= Y_4.\\end{{align*}}\]

The Jacobian determinant of the transformation is calculated.\n\[\begin{{vmatrix}}\frac{{\partial X_1}}{{\partial Y_1}} & \frac{{\partial X_1}}{{\partial Y_2}} & \frac{{\partial X_1}}{{\partial Y_3}} & \frac{{\partial X_1}}{{\partial Y_4}}\\\frac{{\partial X_2}}{{\partial Y_1}} & \frac{{\partial X_2}}{{\partial Y_2}} & \frac{{\partial X_2}}{{\partial Y_3}} & \frac{{\partial X_2}}{{\partial Y_4}}\\\frac{{\partial X_3}}{{\partial Y_1}} & \frac{{\partial X_3}}{{\partial Y_2}} & \frac{{\partial X_3}}{{\partial Y_3}} & \frac{{\partial X_3}}{{\partial Y_4}}\\\frac{{\partial X_4}}{{\partial Y_1}} & \frac{{\partial X_4}}{{\partial Y_2}} & \frac{{\partial X_4}}{{\partial Y_3}} & \frac{{\partial X_4}}{{\partial Y_4}}\\\end{{vmatrix}}\]This yields \(J = 24 Y_4^3.\)

The joint density of the \(Y_i\) is computed by transforming the density of the \(X_i\), multiplying by the Jacobian determinant, i.e.\n\[f_Y(y_1, y_2, y_3, y_4) = f_X(x_1, x_2, x_3, x_4)|J| = 24(24y_4^3).\]

To show that the \(Y_i\) are independent, it must be shown that the joint density factors as the product of the marginals. Here,\n\[f_Y(y_1, y_2, y_3, y_4) = f_{Y_1}(y_1) f_{Y_2}(y_2) f_{Y_3}(y_3) f_{Y_4}(y_4).\]