Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2016 |
| Nomor Soal | : | 4 |
SOAL
Jika diketahui force of mortality adalah \(\mu _x^{\left( d \right)} = \frac{4}{{5\left( {100 – x} \right)}}\) dan force of withdrawal adalah \(\mu _x^{\left( w \right)} = \frac{6}{{5\left( {100 – x} \right)}}\), hitunglah conditional density function untuk kematian seseorang pada umur \(70 + t\), jika orang tersebut hidup pada umur 70.
- \(\frac{{30 – t}}{{600}}\)
- \(\frac{{30 – t}}{{1200}}\)
- \(\frac{{30 – t}}{{1125}}\)
- \(\frac{{70 – t}}{{1200}}\)
- \(\frac{{70 – t}}{{1800}}\)
| Diketahui | - \(\mu _x^{\left( d \right)} = \frac{4}{{5\left( {100 – x} \right)}}\)
- \(\mu _x^{\left( w \right)} = \frac{6}{{5\left( {100 – x} \right)}}\)
- Kondisional jika orang tersebut hidup pada umur 70
|
| Rumus yang digunakan | - \(\mu _x^{\left( \tau \right)} = \mu _x^{\left( d \right)} + \mu _x^{\left( w \right)}\)
- \({}_tp_x^{\left( \tau \right)} = \exp \left( { – \int\limits_0^t {\mu _x^{\left( \tau \right)}\left( s \right)ds} } \right)\)
- \(S\left( x \right) = {}_x{p_0}\)
|
| Proses pengerjaan | \(\mu _x^{\left( \tau \right)} = \mu _x^{\left( d \right)} + \mu _x^{\left( w \right)}\)
\(\mu _x^{\left( \tau \right)} = \frac{4}{{5\left( {100 – x} \right)}} + \frac{6}{{5\left( {100 – x} \right)}}\)
\(\mu _x^{\left( \tau \right)} = \frac{{10}}{{5\left( {100 – x} \right)}}\)\({}_tp_x^{\left( \tau \right)} = \exp \left( { – \int\limits_0^t {\frac{2}{{\left( {100 – s} \right)}}ds} } \right),{\rm{ Misal }}u = 100 – s \to du = – ds\)
\({}_tp_x^{\left( \tau \right)} = \exp \left( {2\int\limits_{100}^{100 – t} {\frac{1}{u}du} } \right)\)
\({}_tp_x^{\left( \tau \right)} = \exp \left( {2\ln \left( {100 – t} \right) – 2\ln \left( {100} \right)} \right)\)
\({}_tp_x^{\left( \tau \right)} = \frac{{{{\left( {100 – t} \right)}^2}}}{{{{100}^2}}}\)
\(f\left( {t,j} \right) = \frac{{{{\left( {100 – t} \right)}^2}}}{{{{100}^2}}} \cdot \frac{4}{{5\left( {100 – t} \right)}}\)
\(f\left( {t,j} \right) = \frac{{4\left( {100 – t} \right)}}{{5 \cdot {{100}^2}}}\)
\(Peluangnya = \frac{{f\left( {t,j} \right)}}{{S\left( x \right)}}\)
\(Peluangnya = \frac{{{}_tp_{70}^{\left( \tau \right)} \cdot \mu _{70}^{\left( d \right)}}}{{S\left( {70} \right)}}\)
\(Peluangnya = \frac{{\frac{{4\left( {100 – 70 – t} \right)}}{{5 \cdot {{100}^2}}}}}{{\frac{{{{\left( {100 – 70} \right)}^2}}}{{{{100}^2}}}}}\)
\(Peluangnya = \frac{{4\left( {30 – t} \right)}}{5} \cdot \frac{1}{{900}}\)
\(Peluangnya = \frac{{30 – t}}{{1125}}\) |
| Jawaban | C. \(\frac{{30 – t}}{{1125}}\) |
