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 |