Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2016 |
| Nomor Soal | : | 18 |
SOAL
Jika \(\mu _{50 + t}^{(w)}{\rm{ }}\) dan \(\mu _{50 + t}^{(d)}\) bernilai constant pada \(0 < t < 1\), hitunglah \(q_{50}^{(d)}\) jika diketahui \(q_{50}^{‘(d)} = q_{50}^{‘(w)} = 0.4\)
- 0.18
- 0.215
- 0.255
- 0.285
- 0.32
| Diketahui | constant force
\(q_{50}^{‘(d)} = q_{50}^{‘(w)} = 0.4\) |
| Rumus yang digunakan | \(p_x^{(\tau )} = p_x^{‘(w)} \cdot p_x^{‘(d)}\)
\(p_x^{‘(d)} = {\left( {p_x^{(\tau )}} \right)^{p_x^{(d)}/q_x^{(\tau )}}}\) |
| Proses pengerjaan | \(p_{50}^{(\tau )} = p_{50}^{‘(w)} \cdot p_{50}^{‘(d)}\)
\(= \left( {1 – q_{50}^{‘(w)}} \right) \cdot \left( {1 – q_{50}^{‘(d)}} \right)\)
\(= \left( {1 – 0.4} \right)\left( {1 – 0.4} \right)\)
\(= 0.36\)
\(q_{50}^{(\tau )} = 1 – 0.36\)
\(= 0.64\)
\(p_{50}^{‘(d)} = {\left( {p_{50}^{(\tau )}} \right)^{p_{50}^{(d)}/q_{50}^{(\tau )}}}\)
\(\Leftrightarrow \ln \left( {p_{50}^{‘(d)}} \right) = \frac{{p_{50}^{(d)}}}{{q_{50}^{(\tau )}}}\ln \left( {p_{50}^{(\tau )}} \right)\)
\(\Leftrightarrow p_{50}^{(d)} = q_{50}^{(\tau )}\frac{{\ln \left( {p_{50}^{‘(d)}} \right)}}{{\ln \left( {p_{50}^{(\tau )}} \right)}}\)
\(\Leftrightarrow p_{50}^{(d)} = \left( {0.64} \right)\frac{{\ln \left( {0.6} \right)}}{{\ln \left( {0.36} \right)}}\)
\(\Leftrightarrow p_{50}^{(d)} = 0.32\) |
| Jawaban | e. 0.32 |
