Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
5 |
SOAL
Tentukan \({g_{(s)}}\) sehingga \(_s{q_{xy}} = s \cdot {q_{xy}} + {g_{(s)}} \cdot {q_{\bar x\bar y}}\) terpenuhi untuk \(0 \le s \le 1\)
- \({g_{(s)}} = {s^2} – s\)
- \({g_{(s)}} = \sqrt {1 – {s^2}} \)
- \({g_{(s)}} = s\left( {1 – s} \right)\)
- \({g_{(s)}} = \frac{s}{{\sqrt {1 – s} }}\)
- \({g_{(s)}} = 1 – s\)
| Diketahui |
- \(_s{q_{xy}} = s \cdot {q_{xy}} + {g_{\left( s \right)}}\left( {{q_{\bar x\bar y}}} \right)\)
- \(0 \le s \le 1\)
|
| Pembahasan |
\(_s{q_{xy}} = 1 – {\,_s}{p_{xy}}\)
\(= 1 – {\,_s}{p_x}{ \cdot _s}{p_y}\)
\(= 1 – \left( {1{ – _s}{q_x}} \right)\left( {1{ – _s}{q_y}} \right)\)
\(\mathop = \limits^{uniform} s\left( {{q_x} + {q_y}} \right) – {s^2}\left( {{q_x} \cdot {q_y}} \right)\)
\(= s\left( {{q_{xy}} + {q_{\bar x\bar y}}} \right) – {s^2} \cdot {q_{\bar x\bar y}}\)
\(= s \cdot {q_{xy}} – {q_{\bar x\bar y}}\left( {{s^2} – s} \right)\)
\(= s \cdot {q_{xy}} + {q_{\bar x\bar y}}\left( {s – {s^2}} \right)\)
|
|
\({g_{\left( s \right)}} = \left( {s – {s^2}} \right) = s\left( {1 – s} \right)\) |
| Jawaban |
c. \({g_{(s)}} = s\left( {1 – s} \right)\) |
