Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 7 |
SOAL
Jika diketahui force of mortality adalah \(\mu _x^{\left( d \right)} = \frac{4}{{5\left( {100 – x} \right)}}\) dan force of withdrawal adalah \(\mu _x^{\left( w \right)} = \frac{{11}}{{5\left( {100 – x} \right)}}\), hitunglah conditional density function untuk kematian seseorang pada umur \(70 + t\), jika orang tersebut hidup pada umur 70.
- \(\frac{{30 – t}}{{600}}\)
- \(\frac{{70 – t}}{{1125}}\)
- \(\frac{{{{\left( {30 – t} \right)}^2}}}{{1125}}\)
- \(\frac{{{{\left( {70 – t} \right)}^2}}}{{33750}}\)
- \(\frac{{{{\left( {30 – t} \right)}^2}}}{{33750}}\)
| Diketahui | \(\mu _x^{\left( d \right)} = \frac{4}{{5\left( {100 – x} \right)}}\) dan \(\mu _x^{\left( w \right)} = \frac{{11}}{{5\left( {100 – x} \right)}}\)
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 | \(\mu _x^{\left( \tau \right)} = \mu _x^{\left( d \right)} + \mu _x^{\left( w \right)}\)
\(= \frac{4}{{5\left( {100 – x} \right)}} + \frac{{11}}{{5\left( {100 – x} \right)}}\)
\(= \frac{{15}}{{5\left( {100 – x} \right)}}\)
\({}_tp_x^{\left( \tau \right)} = \exp \left( { – \int\limits_0^t {\frac{3}{{\left( {100 – s} \right)}}ds} } \right),\) \({\rm{Misal }}u = 100 – s \to du = – ds\)
\(= \exp \left( {3\int\limits_{100}^{100 – t} {\frac{1}{u}du} } \right)\)
\(= \exp \left( {3\ln \left( {100 – t} \right) – 3\ln \left( {100} \right)} \right)\)
\(= \frac{{{{\left( {100 – t} \right)}^3}}}{{{{100}^3}}}\)
\(f\left( {t,j} \right) = \frac{{{{\left( {100 – t} \right)}^3}}}{{{{100}^3}}} \cdot \frac{4}{{5\left( {100 – t} \right)}}\)
\(= \frac{{4{{\left( {100 – t} \right)}^2}}}{{5 \cdot {{100}^3}}}\)
\(Peluangnya = \frac{{f\left( {t,j} \right)}}{{S\left( x \right)}}\)
\(= \frac{{{}_tp_{70}^{\left( \tau \right)} \cdot \mu _{70}^{\left( d \right)}}}{{S\left( {70} \right)}}\)
\(= \frac{{\frac{{4{{\left( {100 – 70 – t} \right)}^2}}}{{5 \cdot {{100}^3}}}}}{{\frac{{{{\left( {100 – 70} \right)}^3}}}{{{{100}^3}}}}}\)
\(= \frac{{4{{\left( {30 – t} \right)}^2}}}{5} \cdot \frac{1}{{27.000}}\)
\(= \frac{{{{\left( {30 – t} \right)}^2}}}{{33.750}}\) |
| Jawaban | e. \(\frac{{{{\left( {30 – t} \right)}^2}}}{{33750}}\) |
