Pembahasan Soal Ujian Profesi Aktuaris
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 |
|
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| 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 |