Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Diberikan informasi sebagai berikut:
| \(x\) | \({l_x}\) | \({d_x}\) | \({p_x}\) | \({q_x}\) |
| 95 | 0,4 | |||
| 96 | 0,2 | |||
| 97 | 72 | 1,0 |
Jika diketahui \({l_{90}} = 1.000,\,\,\,{l_{93}} = 825\), dan kematian berdistribusi seragam untuk setiap usia, berapakah probabilitas (90) meninggal antara usia 93 dan 95,5?
- 0,123
- 0,234
- 0,345
- 0,456
- 0,567
| Step 1 | \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}{}_{2,5}{q_{93}}\) \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}(1 – {}_{2,5}{p_{93}})\) \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}(1 – {}_2{p_{93}}{}_{0,5}{p_{95}})\) \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}(1 – {}_2{p_{93}}(1 – {}_{0,5}{q_{95}}))\) \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}(1 – {}_2{p_{93}}(1 – 0,5{q_{95}}))\) | ||||||||||||||||||||
| Step 2 | \({}_3{p_{90}} = \frac{{{l_{93}}}}{{{l_{90}}}}\) \({}_3{p_{90}} = \frac{{825}}{{1.000}}\) \({}_3{p_{90}} = 0,825\) | ||||||||||||||||||||
| Step 3 |
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| Step 4 | \({}_2{p_{93}} = \frac{{{l_{95}}}}{{{l_{93}}}}\) \({}_2{p_{93}} = \frac{{600}}{{825}}\) \({}_2{p_{93}} \cong 0,72727\) | ||||||||||||||||||||
| Maka | \({}_{3|2,5}{q_{90}} = {}_3{p_{90}}(1 – {}_2{p_{93}}(1 – 0,5{q_{95}}))\) \({}_{3|2,5}{q_{90}} = 0,825(1 – (0,72727)(1 – 0,5(0,4)))\) \({}_{3|2,5}{q_{90}} = 0,825(0,418184)\) \({}_{3|2,5}{q_{90}} = 0,3450018\) \({}_{3|2,5}{q_{90}} \cong 0,345\) | ||||||||||||||||||||
| Jawaban | c. 0,345 |