The random variables [math]X,Y[/math] have the joint density function

[[math]] f_{X,Y}(x,y) = \begin{cases} c xy \, e^{-xy}, 1 \lt x \lt y \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]

for a constant [math]c[/math]. Determine the joint density function for the random variables [math]W = Y^{-1}, Z = X^{-1}.[/math]

• [[math]] f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-1/wz}, 0 \lt w \lt z \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-w/z}, 0 \lt w \lt z \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-wz}, 1 \lt z \lt w \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c w^2z \, e^{-1/(wz)}, 0\lt z \lt w \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c wz^2 \, e^{-1/(wz)}, 0\lt z \lt w \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]