Pembahasan Soal Ujian Profesi Aktuaris
Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian | : | Matematika Aktuaria |
Periode Ujian | : | Mei 2018 |
Nomor Soal | : | 7 |
SOAL
Diberikan sebuah fungsi survival dari seorang bayi yang baru lahir :
\({S_0}\left( s \right) = \left\{ \begin{array}{l} 1 – \frac{x}{{250}},\,\,0 \le x \le 40\\ 1 – {\left( {\frac{x}{{100}}} \right)^2},\,\,40 \le x \le 100 \end{array} \right.\)
Hitunglah probabilitas seseorang yang berumur 25 tahun akan meninggal dalam 30 tahun.
- 0,210
- 0,215
- 0,220
- 0,225
- 0,230
Diketahui | \({S_0}\left( s \right) = \left\{ \begin{array}{l} 1 – \frac{x}{{250}},\,\,0 \le x \le 40\\ 1 – {\left( {\frac{x}{{100}}} \right)^2},\,\,40 \le x \le 100 \end{array} \right.\) |
Ditanya | \(_{30}{q_{25}}\) |
Step 1 |
\(_{30}{q_{25}} = {\,_{15}}{q_{25}} + {\,_{15}}{p_{25}} \cdot {\,_{15}}{q_{40}}\) ………(*) |
Step 2 | - \(_{15}{p_{25}} = \frac{{S\left( {40} \right)}}{{S\left( {25} \right)}} = \frac{{1 – {{\left( {\frac{{40}}{{100}}} \right)}^2}}}{{1 – \left( {\frac{{25}}{{250}}} \right)}} = \frac{{0,84}}{{0,9}} = 0,9333\)
- \(_{15}{q_{25}} = 1{ – _{15}}{p_{25}} = 1 – 0,9333 = 0,0667\)
- \(_{15}{p_{40}} = \frac{{S\left( {55} \right)}}{{S\left( {40} \right)}} = \frac{{1 – {{\left( {\frac{{55}}{{100}}} \right)}^2}}}{{1 – {{\left( {\frac{{40}}{{100}}} \right)}^2}}} = \frac{{0,6975}}{{0,84}} = 0,830357142\)
- \(_{15}{q_{40}} = 1 – 0,830357142 = 0,169642857\)
|
Step 3 | (*)
\(_{30}{q_{25}} = {\,_{15}}{q_{25}} + {\,_{15}}{p_{25}} \cdot {\,_{15}}{q_{40}}\)
\(= 0,0667 + 0,9333 \cdot 0,169642857\)
\(= 0,225\) |
Jawaban | d. 0,225 |