Revision as of 21:48, 15 January 2024 by Admin (Created page with "You are given <math display = "block">{ }_{t} q_{0}=\frac{t^{2}}{10,000}</math> for <math>0 < t < 100</math>. Calculate <math>\stackrel{\circ}{e}_{\text {75:1010 }}</math>. <ul class="mw-excansopts"><li> 6.6</li><li> 7.0</li><li> 7.4</li><li> 7.8</li><li> 8.2</li></ul>")
Jan 15'24
Exercise
You are given
[[math]]{ }_{t} q_{0}=\frac{t^{2}}{10,000}[[/math]]
for [math]0 \lt t \lt 100[/math].
Calculate [math]\stackrel{\circ}{e}_{\text {75:1010 }}[/math].
- 6.6
- 7.0
- 7.4
- 7.8
- 8.2
Jan 15'24
Answer: E
[[math]]
\begin{aligned}
& e_{75: 10}=\int_{t=0}^{t=10}{ }_{t} p_{75} d t \quad \text { where }{ }_{t} p_{x}=\frac{{ }_{t+x} p_{0}}{{ }_{x} p_{0}}=\frac{1-\frac{(t+x)^{2}}{10000}}{1-\frac{x^{2}}{10000}}=\frac{10000-(t+x)^{2}}{10000-x^{2}} \text { for } 0\lt t \lt100-x \\
& =\int_{0}^{10} \frac{10000-75^{2}-150 t-t^{2}}{10000-75^{2}} d t \\
& =\frac{1}{4375} \cdot\left[4375 t-75 t^{2}-\frac{t^{3}}{3}\right]_{t=0}^{t=10}=8.21
\end{aligned}
[[/math]]
Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.