Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 3 |
SOAL
Diketahui:
- \({S_0}\left( t \right) = {\left( {1 – \frac{t}{\omega }} \right)^{\frac{1}{4}}}\), \({\rm{untuk }}0 \le t \le \omega \)
- \({\mu _{65}} = \frac{1}{{180}}\)
Hitunglah \({e_{106}}\), yaitu ekspektasi hidup pada umur 106.
- 2,48
- 2,59
- 2,70
- 2,81
- 2,92
| Diketahui | - \({S_0}\left( t \right) = {\left( {1 – \frac{t}{\omega }} \right)^{\frac{1}{4}}} \), \({\rm{untuk }}0 \le t \le \omega \)
- \({\mu _{65}} = \frac{1}{{180}}\)
|
| Rumus yang digunakan | \({\mu _x} = \frac{{ – \frac{d}{{dt}}{S_x}\left( t \right)}}{{{S_x}\left( t \right)}}\)
\({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\)
\({e_x} = \sum\limits_{k = 1}^\infty {_k{p_x}} \) |
| Proses Pengerjaan | \({\mu _x} = \frac{{ – \frac{d}{{dt}}{S_0}\left( t \right)}}{{{S_0}\left( t \right)}}\)
\(= \frac{{ – \frac{d}{{dt}}{{\left( {1 – \frac{t}{\omega }} \right)}^{\frac{1}{4}}}}}{{{{\left( {1 – \frac{t}{\omega }} \right)}^{\frac{1}{4}}}}}\)
\(= \frac{{\frac{1}{{4\omega {{\left( {1 – \frac{t}{\omega }} \right)}^{\frac{3}{4}}}}}}}{{{{\left( {1 – \frac{t}{\omega }} \right)}^{\frac{1}{4}}}}}\)
\({\mu _{65}} = \frac{1}{{4\omega \left( {1 – \frac{{65}}{\omega }} \right)}}\)
\(\frac{1}{{180}} = \frac{1}{{4\omega – 260}}\)
\(\omega = \frac{{180 + 260}}{4} = 110\) |
| \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\)
\(= {\left( {\frac{{110 – x – t}}{{110 – x}}} \right)^{\frac{1}{4}}}\) |
| \({e_{106}} = \sum\limits_{k = 1}^4 {_k{p_{106}}} \)
\(= \sum\limits_{k = 1}^4 {{{\left( {\frac{{4 – k}}{4}} \right)}^{\frac{1}{4}}}} \)
\(= 2,4786\) |
| Jawaban | a. 2,48 |
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