Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2015 |
| Nomor Soal | : | 13 |
SOAL
Dalam sebuah studi mortalita, diketahui data sebagai berikut:
Waktu
\({t_i}\) | Jumlah Kematian
\({d_i}\) | Jumlah Risiko
\({Y_i}\) |
| 5 | 2 | 15 |
| 7 | 2 | 12 |
| 10 | 1 | 10 |
| 12 | 2 | 6 |
Hitunglah \(\tilde S\left( {12} \right)\) berdasarkan estimasi Nelson-Aalen \(\tilde H\left( {12} \right)\) (dibulatkan 3 desimal)
- 0,338
- 0,480
- 0,386
- 0,522
- 0,627
| Diketahui | Waktu
\({t_i}\) | Jumlah Kematian
\({d_i}\) | Jumlah Risiko
\({Y_i}\) | | 5 | 2 | 15 | | 7 | 2 | 12 | | 10 | 1 | 10 | | 12 | 2 | 6 |
|
| Rumus yang digunakan | \(\tilde H\left( t \right) = \hat \Lambda \left( t \right) = \sum\limits_{j = 1}^m {\frac{{{d_j}}}{{{r_j}}}} , {t_m} \le t < {t_{m + 1}}\)
\(\tilde S\left( t \right) = \exp \left( { – \tilde H\left( t \right)} \right)\)
\(= \exp \left( { – \sum\limits_{j = 1}^m {\frac{{{d_j}}}{{{r_j}}}} } \right), {t_m} \le t < {t_{m + 1}}\) |
| Proses pengerjaan | \(\tilde H\left( {12} \right) = \sum\limits_{j = 1}^4 {\frac{{{d_j}}}{{{r_j}}}} \)
\(= \frac{2}{{15}} + \frac{2}{{12}} + \frac{1}{{10}} + \frac{2}{6}\)
\(= \frac{{11}}{{15}}\) |
| \(\tilde S\left( 4 \right) = \exp \left( { – \tilde H\left( {12} \right)} \right)\)
\(= \exp \left( { – \frac{{11}}{{15}}} \right)\)
\(= 0,480305\) |
| Jawaban | b. 0,480305 |
