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].

1. [[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]]
2. [[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]]
3. [[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]]
4. [[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]]
5. [[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]]