Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | November 2018 |
| Nomor Soal | : | 29 |
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 | | \(x\) | \({l_x}\) | \({d_x}\) | \({p_x}\) | \({q_x}\) | | 95 | 600 | | 0,6 | 0,4 | | 96 | 360 | | 0,2 | 0,8 | | 97 | 72 | 72 | 0 | 1,0 |
- \({p_x} + {q_x} = 1\)
\({p_{97}} = 1 – 1 = 0\)
- \({q_x} = \frac{{{d_x}}}{{{l_x}}}\)
\({l_{97}} = \frac{{72}}{1} = 72\)
- \({p_x} = \frac{{{l_{x + 1}}}}{{{l_x}}}\)
\({l_{96}} = \frac{{72}}{{0,2}} = 360\)
\({l_{95}} = \frac{{360}}{{0,6}} = 600\)
|
| 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 |
