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 | 3 |
| 25 | 2 | 0 |
| 30 | 0 | \({c_{30}}\) |
| 32 | 9 | 0 |
| 40 | 2 | 0 |
\(\frac{{\hat V\left[ {\hat S\left( {35} \right)} \right]}}{{{{\left[ {\hat S\left( {35} \right)} \right]}^2}}} = 0.012947\)
Hitunglah \({c_{30}}\), jumlah yang disensor pada waktu \(t = 30\)
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| Diketahui |
\(\frac{{\hat V\left[ {\hat S\left( {35} \right)} \right]}}{{{{\left[ {\hat S\left( {35} \right)} \right]}^2}}} = 0.012947\) | |||||||||||||||||||||
| 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 | \(\hat V\left[ {\hat S\left( {35} \right)} \right] = {\left[ {\hat S\left( {35} \right)} \right]^2} \cdot \sum\limits_{j = 1}^3 {\left( {\frac{{{d_j}}}{{{r_j}\left( {{r_j} – {d_j}} \right)}}} \right)} \)
\(0,012947 = \left( {\frac{3}{{50\left( {47} \right)}}} \right) + \left( {\frac{2}{{44\left( {42} \right)}}} \right) + \left( {\frac{9}{{\left( {42 – c} \right)\left( {33 – c} \right)}}} \right)\)
\(0,012947 = 0,001277 + 0,001082 + \frac{9}{{1386 – 75c + {c^2}}}\)
\(0,010588 = \frac{9}{{1386 – 75c + {c^2}}}\)
\(1386 – 75c + {c^2} = 850\)
\({c^2} – 75c + 536 = 0\)
\(\left( {c – 67} \right)\left( {c – 8} \right) = 0\) Karena hanya terdapat 50 peserta maka pilih \({c_{30}} = 8\) | |||||||||||||||||||||
| Jawaban | d. 8 |