Pembahasan Ujian PAI: A60 - No. 8 - Juni 2015

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Matematika Aktuaria
Periode Ujian	:	Juni 2015
Nomor Soal	:	8

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

Kunci Jawaban & Pembahasan

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 digunkan	\(\mu \left( x \right) = – \frac{1}{{S\left( x \right)}} \cdot \frac{d}{{dx}}S\left( x \right)\)
Proses pengerajaan	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

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