exercise:41f9b178df: Difference between revisions
From Stochiki
No edit summary |
No edit summary |
||
| (One intermediate revision by the same user not shown) | |||
| Line 9: | Line 9: | ||
</math> | </math> | ||
A deductible equalling the loss in year 1 is applicable in year 2. If the payment in year 2 | A deductible equalling the loss in year 1 is applicable in year 2. If the payment in year 2 equals <math>Y</math>, determine the joint density function for <math>X,Y</math>, given that <math>Y>0</math>. | ||
<ul class="mw-excansopts"> | <ul class="mw-excansopts"> | ||
<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
g(x,y) &= \begin{cases} | |||
\frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | ||
0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
| Line 22: | Line 22: | ||
<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
g(x,y) &= \begin{cases} | |||
\frac{\alpha^2}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | \frac{\alpha^2}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | ||
0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
| Line 30: | Line 30: | ||
<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
g(x,y) &= \begin{cases} | |||
\frac{\alpha^2\theta^{\alpha}}{(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | \frac{\alpha^2\theta^{\alpha}}{(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | ||
0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
| Line 38: | Line 38: | ||
<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
g(x,y) &= \begin{cases} | |||
\frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha}}, y > 0, x > 0 \\ | \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha}}, y > 0, x > 0 \\ | ||
0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
| Line 46: | Line 46: | ||
<li><math display = "block"> | <li><math display = "block"> | ||
\begin{align*} | \begin{align*} | ||
g(x,y) &= \begin{cases} | |||
\frac{\alpha\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | \frac{\alpha\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y > 0, x > 0 \\ | ||
0, \, \textrm{Otherwise} | 0, \, \textrm{Otherwise} | ||
Latest revision as of 14:06, 15 July 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 payment in year 2 equals [math]Y[/math], determine the joint density function for [math]X,Y[/math], given that [math]Y\gt0[/math].
- [[math]] \begin{align*} g(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*} g(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*} g(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*} g(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*} g(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]]