Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
Juni 2015 |
| Nomor Soal |
: |
14 |
SOAL
Diberikan sebagai berikut:
\({\mu _x} = \left\{ \begin{array}{l} \begin{array}{*{20}{c}} {0,05;}&{50 \le x < 60} \end{array}\\ \begin{array}{*{20}{c}} {0,04;}&{60 \le x < 70} \end{array} \end{array} \right.\)
Hitunglah \({}_{\left. 4 \right|14}{q_{50}}\)
- 0,38
- 0,39
- 0,41
- 0,43
- 0,44
| Diketahui |
\({\mu _x} = \left\{ \begin{array}{l} \begin{array}{*{20}{c}} {0,05;}&{50 \le x < 60} \end{array}\\ \begin{array}{*{20}{c}} {0,04;}&{60 \le x < 70} \end{array} \end{array} \right.\) |
| Rumus yang digunakan |
\({}_{\left. t \right|u}{q_x} = {}_t{p_x} \cdot {}_u{q_{x + t}}\)
\({}_t{q_x} = 1 – {}_t{p_x} = 1 – \exp \left[ { – \int\limits_0^t {{\mu _{x + s}}ds} } \right]\) |
| Proses pengerjaan |
Untuk \(x = 50\) maka \({\mu _{50}} = 0.05\)
\({}_4{p_{50}} = \exp \left[ { – \int\limits_0^4 {0.05ds} } \right]\)
\({}_4{p_{50}} = \exp \left[ { – 0.05\left( 4 \right)} \right]\)
\({}_4{p_{50}} = 0.818731\) |
|
Karena 54 + 14 = 68 maka dipecah
\({\mu _x} = \left\{ \begin{array}{l} \begin{array}{*{20}{c}} {0.05;}&{54 \le x < 60} \end{array}\\ \begin{array}{*{20}{c}} {0.04;}&{60 \le x < 68} \end{array} \end{array} \right.\)
sehingga
\({}_{14}{q_{54}} = 1 – {}_{14}{p_{54}} = 1 – \exp \left[ { – \left( {\int\limits_0^6 {0.05ds} + \int\limits_6^{14} {0.04ds} } \right)} \right]\)
\({}_{14}{q_{54}} = 1 – \exp \left[ { – 0.05\left( 6 \right) – 0.04\left( 8 \right)} \right]\)
\({}_{14}{q_{54}} = 0.462056\) |
|
\({}_{\left. 4 \right|14}{q_{50}} = {}_4{p_{50}} \cdot {}_{14}{q_{54}}\)
\({}_{\left. 4 \right|14}{q_{50}} = \left( {0.818731} \right)\left( {0.462056} \right)\)
\({}_{\left. 4 \right|14}{q_{50}} = 0.3782996\) |
| Jawaban |
a. 0,38 |
