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 |