Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
14 |
SOAL
Diberikan
\({\lambda _X}\left( x \right) = k,\) \(x > 0\)
Tentukanlah \({m_a}\), tingkat kematian central dalam interval \(\left( {a,a + 1} \right)\)
- \(k\)
- \(\frac{k}{2}\)
- \(\frac{k}{{2a}}\)
- \(\frac{{2k}}{{2a + 1}}\)
- 0
| Diketahui |
\({\lambda _X}\left( x \right) = k,\) \(x > 0\)
\({m_a}\) dengan interval \(\left( {a,a + 1} \right)\) |
| Rumus yang digunakan |
\(S\left( x \right) = \exp \left( { – \int\limits_0^x {\lambda \left( y \right)dy} } \right)\)
\({m_a} = \frac{{\int\limits_a^{a + 1} {S\left( y \right)\lambda \left( y \right)} dy}}{{\int\limits_a^{a + 1} {S\left( y \right)} dy}}\) |
| Proses pengerjaan |
\(S\left( x \right) = \exp \left( { – \int\limits_0^x {\lambda \left( y \right)dy} } \right)\)
\(= \exp \left( { – \int\limits_0^x {kdy} } \right)\)
\(= \exp \left( { – kx} \right)\) |
| \(\int\limits_a^{a + 1} {S\left( y \right)\lambda \left( y \right)} dy = \int\limits_a^{a + 1} {{e^{ – ky}} \cdot k} dy\), misal \(u = – ky\) maka \(du = – kdy\)
\(= k\left[ {\int\limits_{ – ka}^{ – k\left( {a + 1} \right)} { – \frac{{{e^u}}}{k}du} } \right]\)
\(= – \int\limits_{ – ka}^{ – k\left( {a + 1} \right)} {{e^u}du} \)
\(= – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}}\) |
| \(\int\limits_a^{a + 1} {S\left( y \right)} dy = \int\limits_a^{a + 1} {{e^{ – ky}}} dy\), misal \(u = – ky\) maka \(du = – kdy\)
\(= \int\limits_{ – ka}^{ – k\left( {a + 1} \right)} { – \frac{{{e^u}}}{k}du} \)
\(= \frac{{ – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}}}}{k}\) |
| \({m_a} = \frac{{\int\limits_a^{a + 1} {S\left( y \right)\lambda \left( y \right)} dy}}{{\int\limits_a^{a + 1} {S\left( y \right)} dy}} = \frac{{ – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}}}}{{\frac{{ – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}}}}{k}}}\)
\(= \frac{{ – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}} \cdot k}}{{ – {e^{ – k\left( {a + 1} \right)}} + {e^{ – ka}}}}\)
\(= k\) |
| Jawaban |
a. \(k\) |
