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 \(E\left[ X \right]\)
- 60
- 65
- 70
- 75
- Tidak ada jawaban yang benar
| 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 |
