exercise:D0665476ff: Difference between revisions
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for a constant <math>c</math>. Determine the joint density function for the random variables <math>W = Y^{-1}, Z = X^{-1}.</math> | for a constant <math>c</math>. Determine the joint density function for the random variables <math>W = Y^{-1}, Z = X^{-1}.</math> | ||
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<li><math display = "block"> | <li><math display = "block"> | ||
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0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
\end{cases}</math></li> | \end{cases}</math></li> | ||
</ | </ul> | ||
Latest revision as of 20:20, 17 March 2024
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]]