Pembahasan Ujian PAI: A60 – No. 7 – Mei 2018

Pembahasan Soal Ujian Profesi Aktuaris

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.

1. 0,210
2. 0,215
3. 0,220
4. 0,225
5. 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