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ABy Admin
Jun 02'22
Exercise
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]]