Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
November 2015 |
| Nomor Soal |
: |
30 |
SOAL
Berdasarkan soal nomor 27. Hitunglah peluang, dengan menggunakan fungsi survival di atas, orang yang berusia \(\left( {57} \right)\) meninggal antara usia 84 dan 100 (pembulatan terdekat)
- 0,11
- 0,15
- 0,16
- 0,18
- 0,19
| Diketahui |
\(\begin{array}{*{20}{c}} {{S_0}\left( t \right) = \frac{{{{\left( {121 – t} \right)}^{\frac{1}{2}}}}}{{11}};}&{t \in \left[ {0,121} \right]} \end{array}\) |
| Rumus yang digunakan |
\({}_t{p_x} = \frac{{{S_0}\left( {x + t} \right)}}{{{S_0}\left( x \right)}}\)
\({}_{\left. t \right|u}{q_x} = {}_t{p_x} \cdot {}_u{q_{x + t}} = {}_t{p_x} – {}_{t + u}{p_x}\) |
| Proses pengerjaan |
\({}_t{p_x} = \frac{{{S_0}\left( {x + t} \right)}}{{{S_0}\left( x \right)}} = \frac{{{{\left( {121 – x – t} \right)}^{\frac{1}{2}}}}}{{11}} \cdot \frac{{11}}{{{{\left( {121 – x} \right)}^{\frac{1}{2}}}}}\)
\({}_{27}{p_{57}} = \frac{{{{\left( {121 – 57 – 27} \right)}^{\frac{1}{2}}}}}{{11}} \cdot \frac{{11}}{{{{\left( {121 – 57} \right)}^{\frac{1}{2}}}}}\)
\({}_{27}{p_{57}} = 0.760345\) |
|
\({}_t{p_x} = \frac{{{S_0}\left( {x + t} \right)}}{{{S_0}\left( x \right)}} = \frac{{{{\left( {121 – x – t} \right)}^{\frac{1}{2}}}}}{{11}} \cdot \frac{{11}}{{{{\left( {121 – x} \right)}^{\frac{1}{2}}}}}\)
\({}_{43}{p_{57}} = \frac{{{{\left( {121 – 57 – 43} \right)}^{\frac{1}{2}}}}}{{11}} \cdot \frac{{11}}{{{{\left( {121 – 57} \right)}^{\frac{1}{2}}}}}\)
\({}_{43}{p_{57}} = 0.572822\) |
|
\({}_{\left. {27} \right|16}{q_{57}} = {}_{27}{p_{57}} – {}_{43}{p_{57}}\)
\({}_{\left. {27} \right|16}{q_{57}} = 0.760345 – 0.572822\)
\({}_{\left. {27} \right|16}{q_{57}} = 0.187523\) |
| Jawaban |
e. 0,19 |
