Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Matematika Aktuaria |
Periode Ujian |
: |
November 2015 |
Nomor Soal |
: |
13 |
SOAL
Diberikan suatu fungsi survival \({S_0}\left( x \right)\), dimana:
\({{S_0}\left( x \right) = 1;}\) \({0 \le x < 1}\)
\({{S_0}\left( x \right) = 1 – \frac{{{e^x}}}{{100}};}\) \({1 \le x < 4,5}\)
\({{S_0}\left( x \right) = 0;}\) \({4,5 \le x}\)
Hitunglah nilai dari \({\mu _4}\) (pembulatan terdekat)
- 0,45
- 0,55
- 0,80
- 1,00
- 1,20
Diketahui |
\({{S_0}\left( x \right) = 1;}\) \({0 \le x < 1}\)
\({{S_0}\left( x \right) = 1 – \frac{{{e^x}}}{{100}};}\) \({1 \le x < 4,5}\)
\({{S_0}\left( x \right) = 0;}\) \({4,5 \le x}\) |
Rumus yang digunakan |
\(\mu \left( x \right) = – \frac{1}{{S\left( x \right)}} \cdot \frac{d}{{dx}}S\left( x \right)\) |
Proses pengerjaan |
Untuk \(t = 4\) maka \({S_0}\left( x \right) = 1 – \frac{{{e^x}}}{{100}}\)
\(\mu \left( x \right) = – \frac{1}{{S\left( x \right)}} \cdot \frac{d}{{dx}}S\left( x \right) = – \frac{1}{{1 – \frac{{{e^x}}}{{100}}}} \cdot \frac{d}{{dx}}\left( {1 – \frac{{{e^x}}}{{100}}} \right)\)
\(\mu \left( x \right) = \frac{{100}}{{100 – {e^x}}} \cdot \frac{{{e^x}}}{{100}}\)
\(\mu \left( 4 \right) = \frac{{100}}{{100 – {e^4}}} \cdot \frac{{{e^4}}}{{100}}\)
\(\mu \left( 4 \right) = 1.202553\) |
Jawaban |
e. 1,20 |