Pembahasan Ujian PAI: A50 – No. 16 – November 2014

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Metoda Statistika
Periode Ujian	:	November 2014
Nomor Soal	:	16

SOAL

Berdasarkan soal nomor 15. Tentukan \(E\left[ X \right]\)

1. 60
2. 65
3. 70
4. 75
5. Tidak ada jawaban yang benar

[showhide type more_text=”Kunci Jawaban & Pembahasan” less_text=”Sembunyikan Kunci Jawaban & Pembahasan”]

Diketahui	\({l_x} = 2.500{\left( {64 – 0,8x} \right)^{\frac{1}{3}}},0 \le x \le 80\) \({}_x{p_0} = \frac{{{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}{4}\) , \({\mu _x} = \frac{4}{{15\left( {64 – 0.8x} \right)}}\) , dan \(f\left( x \right) = \frac{1}{{15}}{\left( {64 – 0.8x} \right)^{ – \frac{2}{3}}}\)
Rumus yang digunakan	\(E\left[ X \right] = \int\limits_0^\infty {x \cdot f\left( x \right)dx} = \int\limits_0^\infty {x \cdot {}_x{p_0} \cdot {\mu _x}dx} = \int\limits_0^\infty {{}_x{p_0}dx} \)
Proses pengerjaan	\(E\left[ X \right] = \int\limits_0^{80} {{}_x{p_0}dx} = \int\limits_0^{80} {\frac{{{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}}}{4}dx} \) \({E\left[ X \right] = \frac{1}{4}\int\limits_0^{80} {{{\left( {64 – 0.8x} \right)}^{\frac{1}{3}}}dx} }\)  \({{\rm{misal\_}}{u^3} = 64 – 0.8x \Rightarrow 3{u^2}du = – 0.8dx}\) \(E\left[ X \right] = \frac{1}{4}\int\limits_4^0 {u \cdot \left( { – \frac{{3{u^2}}}{{0.8}}} \right)du} \) \(E\left[ X \right] = – \frac{{3.75}}{4}\int\limits_4^0 {{u^3}du} \) \(E\left[ X \right] = – \frac{{3.75}}{4}\left[ {0 – \frac{{{4^4}}}{4}} \right]\) \(E\left[ X \right] = 60\)
Jawaban	A. 60

[/showhide]