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 |