Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2015 |
| Nomor Soal | : | 14 |
SOAL
Data di bawah ini diekstrak dari table mortalita select dan ultimate dengan periode seleksi 2 tahun:
| \(x\) | \({l_{\left[ x \right]}}\) | \({l_{\left[ x \right] + 1}}\) | \({l_{x + 2}}\) | \(x + 2\) |
| 60 | 80.625 | 79.954 | 78.839 | 62 |
| 61 | 79.137 | 78.402 | 77.252 | 63 |
| 62 | 77.575 | 76.770 | 75.578 | 64 |
Hitunglah \({}_{0,9}{q_{\left[ {60} \right] + 0,6}}\) (dibulatkan 4 desimal)
- 0,0102
- 0,0103
- 0,0104
- 0,0105
- 0,0106
| Diketahui | | \(x\) | \({l_{\left[ x \right]}}\) | \({l_{\left[ x \right] + 1}}\) | \({l_{x + 2}}\) | \(x + 2\) | | 60 | 80.625 | 79.954 | 78.839 | 62 | | 61 | 79.137 | 78.402 | 77.252 | 63 | | 62 | 77.575 | 76.770 | 75.578 | 64 | |
| Rumus yang digunakan | \({q_x} = \frac{{{l_x} – {l_{x + 1}}}}{{{l_x}}},{\rm{ }}{q_{\left[ x \right] + 1}} = \frac{{{l_{\left[ x \right] + 1}} – {l_{x + 2}}}}{{{l_{\left[ x \right] + 1}}}},{\rm{ }}{q_{\left[ x \right]}} = \frac{{{l_{\left[ x \right]}} – {l_{\left[ x \right] + 1}}}}{{{l_{\left[ x \right]}}}}\)
\({}_t{p_x} = {}_{t + x}{p_0}\)
\({}_t{p_x} = {p_x}{p_{x + 1}}{p_{x + 2}} \cdots {p_{x + t – 1}}\)
Untuk UUD konstan dan usia bukan bilangan bulat
\({}_s{p_x} = 1 – s \cdot {q_x}\)
\({}_s{p_{x + t}} = \frac{{{}_{s + t}{p_x}}}{{{}_t{p_x}}} = \frac{{{}_{s + t}{p_x}}}{{1 – t \cdot {q_x}}}\) |
| Proses pengerjaan | \({}_{0,9}{q_{\left[ {60} \right] + 0,6}} = 1 – {}_{0,9}{p_{\left[ {60} \right] + 0,6}}\)
\(= 1 – \frac{{{}_{1,5}{p_{\left[ {60} \right]}}}}{{{}_{0,6}{p_{\left[ {60} \right]}}}}\)
\(= 1 – \frac{{{p_{\left[ {60} \right]}} \cdot {}_{0,5}{p_{\left[ {60} \right] + 1}}}}{{{}_{0,6}{p_{\left[ {60} \right]}}}}\)
\(= 1 – \frac{{{p_{\left[ {60} \right]}} \cdot \left( {1 – 0,5 \cdot {q_{\left[ {60} \right] + 1}}} \right)}}{{\left( {1 – 0,6 \cdot {q_{\left[ {60} \right]}}} \right)}}\)
\(= 1 – \frac{{\frac{{79.954}}{{80.625}} \cdot \left( {1 – 0,5 \cdot \frac{{79.954 – 78.839}}{{79.954}}} \right)}}{{\left( {1 – 0,6 \cdot \frac{{80.625 – 79.954}}{{80.625}}} \right)}}\)
\(= 0,010295\) |
| Jawaban | b. 0,0103 |
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