Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | November 2016 |
| Nomor Soal | : | 6 |
SOAL
Untuk “two independent lives now” usia 30 dan 34, diberikan sebagai berikut:
| \((x)\) | \({q_x}\) |
| 30 | 0,1 |
| 31 | 0,2 |
| 32 | 0,3 |
| 33 | 0,4 |
| 34 | 0,5 |
| 35 | 0,6 |
| 36 | 0,7 |
| 37 | 0,8 |
Hitunglah peluang dimana kematian terakhir dari “two lives” ini akan terjadi selama 3 tahun dari sekarang \(\left( {{}_2|{q_{\overline {30:34} }}} \right)\)
- 0,01
- 0,03
- 0,14
- 0,18
- 0,24
| Rumus | \({}_t|{q_{\overline {x:y} }} = {}_{t + 1}{q_{\overline {x:y} }} – {}_t{q_{\overline {x:y} }}\) |
| Step 1 | \({}_2|{q_{\overline {30:34} }} = {}_3{q_{\overline {30:34} }} – {}_2{q_{\overline {30:34} }}\)
\({}_2|{q_{\overline {30:34} }} = \left( {{}_3{q_{30}}} \right)\left( {{}_3{q_{34}}} \right) – \left( {{}_2{q_{30}}} \right)\left( {{}_2{q_{34}}} \right)\)
\({}_2|{q_{\overline {30:34} }} = \left( {1 – {}_3{p_{30}}} \right)\left( {1 – {}_3{p_{34}}} \right) – \left( {1 – {}_2{p_{30}}} \right)\left( {1 – {}_2{p_{34}}} \right)\)
\({}_2|{q_{\overline {30:34} }} = \left( {1 – \left( {{p_{30}}} \right)\left( {{p_{31}}} \right)\left( {{p_{32}}} \right)} \right)\left( {1 – \left( {{p_{34}}} \right)\left( {{p_{35}}} \right)\left( {{p_{36}}} \right)} \right) – \left( {1 – \left( {{p_{30}}} \right)\left( {{p_{31}}} \right)} \right)\left( {1 – \left( {{p_{34}}} \right)\left( {{p_{35}}} \right)} \right)\)
\({}_2|{q_{\overline {30:34} }} = \left( {0,496} \right)\left( {0,94} \right) – \left( {0,28} \right)\left( {0,8} \right)\)
\({}_2|{q_{\overline {30:34} }} = 0,24224\)
\({}_2|{q_{\overline {30:34} }} \cong 0,24\) |
| Jawaban | e. 0,24 |
