Pembahasan Ujian PAI: A50 - No. 15 - November 2014

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Metoda Statistika
Periode Ujian	:	November 2014
Nomor Soal	:	15

SOAL

Jika diketahui \({l_x} = 2.500{\left( {64 – 0,8x} \right)^{\frac{1}{3}}},0 \le x \le 80\) tentukan \(f\left( x \right)\)

\(\frac{1}{{15}}{\left( {64 – 0,8x} \right)^{\frac{2}{3}}}\)
\(\frac{1}{{15}}{\left( {64 – 0,8x} \right)^{ – \frac{1}{3}}}\)
\(\frac{1}{{15}}{\left( {64 – 0,8x} \right)^{ – \frac{2}{3}}}\)
\(\frac{1}{{15}}{\left( {64 – 0,8x} \right)^{\frac{1}{3}}}\)
Tidak ada jawaban yang benar

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Diketahui	\({l_x} = 2.500{\left( {64 – 0,8x} \right)^{\frac{1}{3}}},0 \le x \le 80\)
Rumus yang digunakan	\(f\left( x \right) = {}_x{p_0} \cdot {\mu _x}\) \({}_x{p_0} = S\left( x \right) = \frac{{{l_x}}}{{{l_0}}}\) \({\mu _x} = \frac{1}{{{l_x}}}\left( { – \frac{d}{{dx}}{l_x}} \right)\)
Proses pengerjaan	\({}_x{p_0} = S\left( x \right) = \frac{{{l_x}}}{{{l_0}}} = \frac{{2,500{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}{{2,500{{\left( {64 – 0.8\left( 0 \right)} \right)}^{\frac{1}{3}}}}} = \frac{{{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}{4}\) \({\mu _x} = \frac{1}{{{l_x}}}\left( { – \frac{d}{{dx}}{l_x}} \right) = \frac{1}{{2,500{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}\left( { – \frac{d}{{dx}}\left( {2,500{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}} \right)} \right)\) \({\mu _x} = \frac{1}{{2,500{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}\left( {\frac{{2,500\left( {0.8} \right)}}{{3{{\left( {64 – 0.8x} \right)}^{\frac{2}{3}}}}}} \right)\) \({\mu _x} = \frac{{0.8}}{{3\left( {64 – 0.8x} \right)}} = \frac{4}{{15\left( {64 – 0.8x} \right)}}\) \(f\left( x \right) = {}_x{p_0} \cdot {\mu _x} = \frac{{{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}{4} \cdot \frac{4}{{15\left( {64 – 0.8x} \right)}}\) \(f\left( x \right) = \frac{1}{{15}}{\left( {64 – 0.8x} \right)^{ – \frac{2}{3}}}\)
Jawaban	C. \(\frac{1}{{15}}{\left( {64 – 0,8x} \right)^{ – \frac{2}{3}}}\)

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