Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Juni 2015 |
| Nomor Soal |
: |
7 |
SOAL
Berdasarkan soal nomor 6
Tentukan \({}_t{p_x}\)
- \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{{x + 1}}{{\left( {x + t} \right)}},}&{x \ge 0,}&{t \ge 0} \end{array}\)
- \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{1}{{{{\left( {x + t + 1} \right)}^2}}},}&{x \ge 0,}&{t \ge 0} \end{array}\)
- \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{1}{{\left( {x + t + 1} \right)}},}&{x \ge 0,}&{t \ge 0} \end{array}\)
- \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{{x + 1}}{{{{\left( {x + t + 1} \right)}^2}}},}&{x \ge 0,}&{t \ge 0} \end{array}\)
- \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{{x + 1}}{{\left( {x + t + 1} \right)}},}&{x \ge 0,}&{t \ge 0} \end{array}\)
| Diketahui |
\(\begin{array}{*{20}{c}} {\mu \left( x \right) = \frac{1}{{x + 1}},}&{x \ge 0} \end{array}\) |
| Rumus yang digunakan |
\({}_t{p_x} = \exp \left[ { – \int\limits_x^{x + t} {\mu \left( s \right)ds} } \right]\) atau \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\) |
| Proses pengerjaan |
Cara 1
\({}_t{p_x} = \exp \left[ { – \int\limits_x^{x + t} {\left( {\frac{1}{{s + 1}}} \right)ds} } \right] = \exp \left[ { – \ln \left( {x + t + 1} \right) + \ln \left( {x + 1} \right)} \right] = \frac{{x + 1}}{{x + t + 1}}\) |
|
Cara 2
\({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}} = \frac{{\frac{1}{{x + t + 1}}}}{{\frac{1}{{x + 1}}}} = \frac{{x + 1}}{{x + t + 1}}\) |
| Jawaban |
e. \(\begin{array}{*{20}{c}} {{}_t{p_x} = \frac{{x + 1}}{{\left( {x + t + 1} \right)}},}&{x \ge 0,}&{t \ge 0} \end{array}\) |
