Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Sebuah studi mortalita dilakukan atas pengamatan terhadap 50 peserta dimulai dari waktu 0. Diketahui:
| Waktu \(t\) | Jumlah Kematian \({d_t}\) | Jumlah yang disensor \({c_t}\) |
| 15 | 3 | 0 |
| 17 | 0 | 4 |
| 25 | 3 | 0 |
| 30 | 0 | \({c_{30}}\) |
| 32 | 8 | 0 |
| 40 | 2 | 0 |
\(\frac{{\hat V\left[ {\hat S\left( {35} \right)} \right]}}{{{{\left[ {\hat S\left( {35} \right)} \right]}^2}}} = 0,012718\)
Hitunglah \({c_{30}}\), jumlah yang disensor pada waktu \(t = 30\)
- 6
- 7
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- 9
- 10
| Diketahui |
\(\frac{{\hat V\left[ {\hat S\left( {35} \right)} \right]}}{{{{\left[ {\hat S\left( {35} \right)} \right]}^2}}} = 0,012718\) | |||||||||||||||||||||
| Rumus yang digunakan | \(\hat V\left[ {\hat S\left( t \right)} \right] = {\left[ {\hat S\left( t \right)} \right]^2} \cdot \sum\limits_{j = 1}^k {\left( {\frac{{{d_j}}}{{{r_j}\left( {{r_j} – {d_j}} \right)}}} \right)} \) | |||||||||||||||||||||
| Proses pengerjaan | \(\hat V\left[ {\hat S\left( {35} \right)} \right] = {\left[ {\hat S\left( {35} \right)} \right]^2} \cdot \sum\limits_{j = 1}^3 {\left( {\frac{{{d_j}}}{{{r_j}\left( {{r_j} – {d_j}} \right)}}} \right)} \) \(0,012718 = \left( {\frac{3}{{50\left( {47} \right)}}} \right) + \left( {\frac{3}{{43\left( {40} \right)}}} \right) + \left( {\frac{8}{{\left( {40 – c} \right)\left( {32 – c} \right)}}} \right)\) \(0,012718 = 0,001277 + 0,001744 + \frac{8}{{1280 – 72c + {c^2}}}\) \(0,009697 = \frac{8}{{1280 – 72c + {c^2}}}\) \(1280 – 72c + {c^2} = 825\) \({c^2} – 72c + 455 = 0\) \(\left( {c – 65} \right)\left( {c – 7} \right) = 0\) Karena hanya terdapat 50 peserta maka pilih \({c_{30}} = 7\) | |||||||||||||||||||||
| Jawaban | b. 7 |