Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
November 2014 |
| Nomor Soal |
: |
15 |
SOAL
Diketahui \({\lambda _X}\left( x \right) = {\left( {80 – x} \right)^{ – \frac{1}{2}}}\) untuk \(0 < x < 80\). Manakah dari nilai di bawah ini yang paling mendekati median dari distribusi \({T_{20}}\)?
- 1,249
- 3,249
- 4,249
- 5,249
- 6,249
| Diketahui |
Force of mortality \({\lambda _X}\left( x \right) = {\left( {80 – x} \right)^{ – \frac{1}{2}}}\) untuk \(0 < x < 80\) |
| Rumus yang digunakan |
\(\Pr \left[ {{T_X} > t} \right] = {}_t{p_x} = \exp \left[ { – \int\limits_0^t {{\mu _{x + s}}ds} } \right]\)
Median: \(\Pr \left[ {{T_x} \le t} \right] = \Pr \left[ {{T_x} > t} \right] = 0.5\) |
| Proses pengerjaan |
\({\mu _{x + s}} = {\left( {80 – x – s} \right)^{ – \frac{1}{2}}}\)
\({}_t{p_{20}} = \exp \left[ { – \int\limits_0^t {{\mu _{x + s}}ds} } \right]\)
\({{}_t{p_{20}} = \exp \left[ { – \int\limits_0^t {\left( {{{\left( {60 – s} \right)}^{ – \frac{1}{2}}}} \right)ds} } \right]}\) misal \({{u^2} = 60 – s \Rightarrow 2udu}\) \(= – ds\)
\({}_t{p_{20}} = \exp \left[ { – \int\limits_{}^{} {\left( {\frac{{ – 2u}}{u}} \right)du} } \right]\)
\({}_t{p_{20}} = \exp \left[ {\int\limits_{\sqrt {60} }^{\sqrt {60 – t} } {2du} } \right]\)
\({}_t{p_{20}} = \exp \left[ {2\sqrt {60 – t} – 2\sqrt {60} } \right]\) |
|
Median
\({}_t{p_{20}} = \exp \left[ {2\sqrt {60 – t} – 2\sqrt {60} } \right] = 0.5\)
\(2\sqrt {60 – t} – 2\sqrt {60} = \ln \left( {0.5} \right)\)
\(t = 60 – {\left[ {\frac{{\ln \left( {0.5} \right) + 2\sqrt {60} }}{2}} \right]^2}\)
\(t = 5.248982\) |
| Jawaban |
d. 5,249 |
