Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
April 2019 |
| Nomor Soal |
: |
21 |
SOAL
Diberikan informasi berikut
- \(e_{30:\overline {40|} }^0 = 27,692\)
- \({S_0}\left( t \right) = 1 – \frac{t}{\omega }\) dimana \(0 \le t \le \omega \)
- \({T_x}\) adalah peubah acak future lifetime untuk \(x\)
Hitunglah \(Var\left( {{T_{30}}} \right)\)
- 332
- 352
- 372
- 392
- 412
| Diketahui |
- \(e_{30:\overline {40|} }^0 = 27,692\)
- \({S_0}\left( t \right) = 1 – \frac{t}{\omega }\) dimana \(0 \le t \le \omega \)
- \({T_x}\) adalah peubah acak future lifetime untuk \(x\)
|
| Rumus yang digunakan |
\(e_{x:\bar n|}^0 = \int\limits_0^n {{}_t{p_x}} dt\), \(e_x^0 = \int\limits_0^\infty {{}_t{p_x}dt} \)
\(Var\left( {{T_x}} \right) = 2\int\limits_0^\infty {t \cdot {}_t{p_x}dt} – {\left( {e_x^0} \right)^2}\) |
| Proses pengerjaan |
Fungsi Survival diasumsikan menggunakan Hukum De Moivre
\(e_{x:\bar n|}^0 = \int\limits_0^n {\frac{{\omega – x – t}}{{\omega – x}}} dt\)
\(e_{30:\overline {40} |}^0 = \int\limits_0^{40} {\frac{{\omega – 30 – t}}{{\omega – 30}}dt} = \int\limits_0^{40} {\left( {1 – \frac{t}{{\omega – 30}}} \right)dt} \)
\(= \left. {t – \frac{{{t^2}}}{{2\left( {\omega – 30} \right)}}} \right|_0^{40}\)
\(27,692 = 40 – \frac{{{{40}^2}}}{{2\left( {\omega – 30} \right)}}\)
\(\omega = \frac{{{{40}^2}}}{{\left( {40 – 27,692} \right)}} + 30\)
\(= 94,998375 \approx 95\) |
|
\(Var\left( {{T_{30}}} \right) = 2\int\limits_0^{95 – 30} {t \cdot {}_t{p_{30}}dt} – {\left( {\int\limits_0^{95 – 30} {{}_t{p_{30}}dt} } \right)^2}\)
\(= 2\int\limits_0^{65} {\frac{{65t – {t^2}}}{{65}}dt} – {\left( {\int\limits_0^{65} {\frac{{65 – t}}{{65}}dt} } \right)^2}\)\(= \frac{{4225}}{3} – \frac{{4225}}{4} = 352,08333\) |
| Jawaban |
b. 352 |
