Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Present value random variable untuk satu polis asuransi milik (x) dapat dinyatakan sebagai:
\(Z = \left\{ \begin{array}{l} 0, & Tx \le 10\\ {v^{Tx}}, & 10 < Tx \le 20\\ 2{v^{Tx}}, & 20 < Tx \le 30\\ 0, & lainnya \end{array} \right.\)
Dari pilihan-pilihan berikut, manakah ekspresi yang tepat untuk menggambarkan \(E[Z]?\)
- \({}_{10}{E_x}[{\bar A_{x + 10}} + {}_{10}{E_{x + 10}}{\bar A_{x + 20}} – {}_{10}{E_{x + 20}}{\bar A_{x + 30}}]\)
- \({}_{10}{E_x}{\bar A_{x + 10}} + {}_{20}{E_x}{\bar A_{x + 20}} – 2{}_{30}{E_x}{\bar A_{x + 30}}\)
- \({}_{10}{E_x}{\bar A_x} + {}_{20}{E_x}{\bar A_{x + 20}} – 2{}_{30}{E_x}{\bar A_{x + 30}}\)
- \({\bar A_x} + {}_{20}{E_x}{\bar A_{x + 20}} – 2{}_{30}{E_x}{\bar A_{x + 30}}\)
- \({}_{10|}{\bar A_x}{ + _{20|}}{\bar A_x}{ – _{30|}}{\bar A_x}\)
| Maka | \(E[Z] = \,\int\limits_{10}^{30} {{v^{{T_x}}}} {f_{Tx}}(t)\,dt + \int\limits_{20}^{30} {{v^{{T_x}}}} {f_{Tx}}(t)\,dt\) \(E[Z]\, = {}_{10}|{}_{20}{\bar A_x} + {}_{20}{|_{10}}{\bar A_x}\) \(E[Z]\, = {}_{10}{E_x}\,{}_{20}{\bar A_{x + 10}} + {}_{20}{E_x}\,{}_{10}{\bar A_{x + 20}}\) \(E[Z]\, = \left( {{}_{10}{E_x}\,{{\bar A}_{x + 10}}\, – {}_{30}{E_x}\,{{\bar A}_{x + 30}}\,} \right) + \left( {{}_{20}{E_x}\,{{\bar A}_{x + 20}}\, – {}_{30}{E_x}\,{{\bar A}_{x + 30}}\,} \right)\) \(E[Z]\, = {}_{10}{E_x}\,{\bar A_{x + 10}}\, + {}_{20}{E_x}\,{\bar A_{x + 20}} – 2{}_{30}{E_x}\,{\bar A_{x + 30}}\) |
| Jawaban | b. \({}_{10}{E_x}{\bar A_{x + 10}} + {}_{20}{E_x}{\bar A_{x + 20}} – 2{}_{30}{E_x}{\bar A_{x + 30}}\) |