91Ó°ÊÓ

We are given the following p.d.f.

\(f\left( {{x_1},{x_2},{x_3}} \right) = \left\{ \begin{array}{l}8{x_1}{x_2}{x_3}\;for\;0 < {x_i} < 1\left( {i = 1,2,3} \right),\\0,otherwise\end{array} \right.\)

The random variables are:

\(\begin{array}{l}{Y_1} = {X_1}\\{Y_2} = {X_1}{X_2}\;\\{Y_3} = {X_1}{X_2}{X_3}.\end{array}\)

The inverse transformations of the variables are:

\(\begin{aligned}{X_1} &= {Y_1}\\{X_2} &= \frac{{{Y_2}}}{{{X_1}}}\\ &= \frac{{{Y_2}}}{{{Y_1}}}\end{aligned}\)

\(\begin{aligned}{X_3} &= \frac{{{Y_3}}}{{{X_1}{X_2}}}\\ &= \frac{{{Y_3}}}{{{Y_2}}}.\end{aligned}\)

Hence, the transformed variables are:

\({X_1} = {Y_1},\)

\({X_2} = \frac{{{Y_2}}}{{{Y_1}}},\)

\({X_3} = \frac{{{Y_3}}}{{{Y_2}}}\)

The joint pdf of \({{\bf{Y}}_{\bf{1}}}{\bf{,}}{{\bf{Y}}_{\bf{2}}}{\bf{,}}{{\bf{Y}}_{\bf{3}}}\) is given as follows,\({\bf{g}}\left( {{{\bf{y}}_{\bf{1}}}{\bf{,}}{{\bf{y}}_{\bf{2}}}{\bf{,}}{{\bf{y}}_{\bf{3}}}} \right){\bf{ = }}\left\{ \begin{array}{l}f\left( {{{\bf{y}}_{\bf{1}}}{\bf{,}}{{\bf{y}}_{\bf{2}}}{\bf{,}}{{\bf{y}}_{\bf{3}}}} \right)\left| {\bf{J}} \right|\;\;\;\;\;{\bf{for}}\;{\bf{0 < }}{{\bf{y}}_{\bf{3}}}{\bf{ < }}{{\bf{y}}_{\bf{2}}}{\bf{ < }}{{\bf{y}}_{\bf{1}}}{\bf{ < 1}}\\{\bf{0}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\bf{otherwise}}\end{array} \right.\)

The jacobian is determined as follows,

\(\begin{aligned}J &= \left| {\begin{array}{*{20}{c}}{\frac{{\partial {x_1}}}{{\partial {y_1}}}}&{\frac{{\partial {x_2}}}{{\partial {y_1}}}}&{\frac{{\partial {x_3}}}{{\partial {y_1}}}}\\{\frac{{\partial {x_1}}}{{\partial {y_2}}}}&{\frac{{\partial {x_2}}}{{\partial {y_2}}}}&{\frac{{\partial {x_3}}}{{\partial {y_2}}}}\\{\frac{{\partial {x_1}}}{{\partial {y_3}}}}&{\frac{{\partial {x_2}}}{{\partial {y_3}}}}&{\frac{{\partial {x_3}}}{{\partial {y_3}}}}\end{array}} \right|\\ &= \left| {\begin{array}{*{20}{c}}1&{\frac{{ - {y_2}}}{{{y_1}^2}}}&0\\0&{\frac{1}{{{y_1}}}}&{\frac{{ - {y_3}}}{{{y_2}^2}}}\\0&0&{\frac{1}{{{y_2}}}}\end{array}} \right|\\ &= \frac{1}{{{y_1}{y_2}}}\;\end{aligned}\)

Now, the Jacobian of the transformed variable:\(\frac{{\bf{1}}}{{{{\bf{y}}_{\bf{1}}}{{\bf{y}}_{\bf{2}}}}}\)

The joint pdf is obtained using the formula as stated below,

\(g\left( {{y_1},{y_2},{y_3}} \right) = f\left( {{x_1},{x_2},{x_3}} \right)\left| J \right|\)

\(\begin{aligned}g\left( {{y_1},{y_2},{y_3}} \right) &= 8\left( {{y_1}} \right)\left( {\frac{{{y_2}}}{{{y_1}}}} \right)\left( {\frac{{{y_3}}}{{{y_2}}}} \right)\left( {\frac{1}{{{y_1}{y_2}}}} \right) \Rightarrow g\left( {{y_1},{y_2},{y_3}} \right)\\ &= \left\{ \begin{array}{l}8{y_3}{\left( {{y_1}{y_2}} \right)^{ - 1}}\;for\;0 < {y_3} < {y_2} < {y_1} < 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\end{aligned}\)