Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Diketahui probabilitas seseorang yang berumur 50 untuk hidup selama \(t\) tahun adalah
\({}_t{p_{50}} = {e^{0.5\left( {1 – {{1.05}^t}} \right)}}\).
Hitunglah \({q_{80}}\)
- 0,06418
- 0,10242
- 0,12804
- 0,18065
- 0,21312
| Diketahui | \({}_t{p_{50}} = {e^{0.5\left( {1 – {{1.05}^t}} \right)}}\) |
| Rumus yang digunakan | \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\) \({}_t{q_x} = 1 – {}_t{p_x}\) |
| Proses pengerjaan | \({q_{80}} = 1 – {p_{80}}\) jadi pertama kita cari \({p_{80}}\) \({p_{80}} = \frac{{S\left( {80 + 1} \right)}}{{S\left( {80} \right)}}\) \(= \frac{{S\left( {50 + 31} \right)}}{{S\left( {50} \right)}} \cdot \frac{{S\left( {50} \right)}}{{S\left( {50 + 30} \right)}}\) \(= \frac{{\frac{{S\left( {50 + 31} \right)}}{{S\left( {50} \right)}}}}{{\frac{{S\left( {50 + 30} \right)}}{{S\left( {50} \right)}}}}\) \(= \frac{{{}_{31}{p_{50}}}}{{{}_{30}{p_{50}}}}\) \(= \frac{{\exp \left( {0.5\left( {1 – {{1.05}^{31}}} \right)} \right)}}{{\exp \left( {0.5\left( {1 – {{1.05}^{30}}} \right)} \right)}}\) \(= 0,897584\) \({q_{80}} = 1 – {p_{80}}\) \(= 1 – 0,897584\) \(= 0,102416\) |
| Jawaban | b. 0,10242 |