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A deductible equalling the loss in year 1 is applicable in year 2. If the loss in year 2, with deductible in effect, equals <math>Y</math>, determine the joint density function for <math>X,Y</math>. | A deductible equalling the loss in year 1 is applicable in year 2. If the loss in year 2, with deductible in effect, equals <math>Y</math>, determine the joint density function for <math>X,Y</math>. | ||
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<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
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\end{align*} | \end{align*} | ||
</math></li> | </math></li> | ||
</ | </ul> | ||
Revision as of 21:16, 17 March 2024
The loss in year 1, [math]X[/math], has probability density function
[[math]]
f(x) = \begin{cases}
\frac{\alpha \theta^{\alpha}}{(x+\theta)^{\alpha +1}}, x
\geq 0 \\
0, x \lt 0
\end{cases}
[[/math]]
A deductible equalling the loss in year 1 is applicable in year 2. If the loss in year 2, with deductible in effect, equals [math]Y[/math], determine the joint density function for [math]X,Y[/math].
- [[math]] \begin{align*} f_{X,Y}(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
- [[math]] \begin{align*} f_{X,Y}(x,y) &= \begin{cases} \frac{\alpha^2}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
- [[math]] \begin{align*} f_{X,Y}(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
- [[math]] \begin{align*} f_{X,Y}(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
- [[math]] \begin{align*} f_{X,Y}(x,y) &= \begin{cases} \frac{\alpha\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]