Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Sebuah survival model didefinisikan sebagai berikut:
\({S_X}\left( x \right) = \frac{{c – x}}{{c + x}}\), untuk \(0 \le x \le c\). Kemudian, sebuah tabel kehidupan (life table) disusun berdasarkan distribusi tersebut dengan radix 100,000. Dalam tabel tersebut, \({l_{35}} = 44,000\). DIketahui pula \(\omega = 90\). Hitunglah probabilitas dari seorang berusia 10 tahun akan meninggal antara usia 30 dan 45.- \(\frac{{11}}{{24}}\)
- \(\frac{9}{{24}}\)
- \(\frac{7}{{24}}\)
- \(\frac{5}{{24}}\)
- \(\frac{1}{8}\)
| Diketahui |
|
| Rumus yang digunakan | \({S_X}\left( x \right) = {}_x{p_0} = \frac{{{l_x}}}{{{l_0}}}\) \({}_{\left. t \right|u}{q_x} = {}_t{p_x} \cdot {}_u{q_{x + t}}\) \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\) |
| Proses pengerjaan | \({S_X}\left( {35} \right) = \frac{{c – 35}}{{c + 35}} = \frac{{{l_{35}}}}{{{l_0}}}\) \(\frac{{44,000}}{{100,000}} = \frac{{c – 35}}{{c + 35}}\) \(44c + 1,540 = 100c – 3,500\) \(c = \frac{{5040}}{{56}}\) \(c = 90\) |
| \({}_{\left. {20} \right|15}{q_{10}} = {}_{20}{p_{10}} \cdot {}_{15}{q_{30}}\) \({}_{\left. {20} \right|15}{q_{10}} = \frac{{S\left( {30} \right)}}{{S\left( {10} \right)}} \cdot \left( {1 – \frac{{S\left( {45} \right)}}{{S\left( {30} \right)}}} \right) = \frac{{S\left( {30} \right)}}{{S\left( {10} \right)}} \cdot \left( {\frac{{S\left( {30} \right) – S\left( {45} \right)}}{{S\left( {30} \right)}}} \right)\) \({}_{\left. {20} \right|15}{q_{10}} = \frac{{S\left( {30} \right) – S\left( {45} \right)}}{{S\left( {10} \right)}}\) \({}_{\left. {20} \right|15}{q_{10}} = \frac{{\frac{{90 – 30}}{{90 + 30}} – \frac{{90 – 45}}{{90 + 45}}}}{{\frac{{90 – 10}}{{90 + 10}}}}\) \({}_{\left. {20} \right|15}{q_{10}} = \frac{5}{{24}}\) | |
| Jawaban | d. \(\frac{5}{{24}}\) |