Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Juni 2016 |
| Nomor Soal | : | 14 |
SOAL
Diketahui
\(F\left( X \right) = 1 – {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\) untuk \(0 \le x \le 120,\)
Hitunglah \(E\left[ T \right]\) untuk \(x = 30\), yaitu \(\mathop {{e_{30}}}\limits^ \circ \)
- 66,73
- 68,92
- 70,17
- 74,63
- 77,14
| Diketahui | \(F\left( X \right) = 1 – {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\) untuk \(0 \le x \le 120,\) |
| Rumus yang digunakan | \(S\left( X \right) = 1 – F\left( X \right)\)
\({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\)
\(\mathop {{e_x}}\limits^ \circ = \int\limits_0^\infty {{}_t{p_x}dt} \) |
| Proses Pengerjaan | \(S\left( X \right) = 1 – F\left( X \right)\)
\(= {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\) |
| \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\)
\(= \frac{{{{\left( {\frac{{120 – x – t}}{{120}}} \right)}^{\frac{1}{6}}}}}{{{{\left( {\frac{{120 – x}}{{120}}} \right)}^{\frac{1}{6}}}}}\)
\(= {\left( {\frac{{120 – x – t}}{{120 – x}}} \right)^{\frac{1}{6}}}\) |
| \(\mathop {{e_{30}}}\limits^ \circ = \int\limits_0^{90} {{}_t{p_{30}}dt} \)
\(= \int\limits_0^{90} {{{\left( {\frac{{90 – t}}{{90}}} \right)}^{\frac{1}{6}}}dt} \)
\(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\int\limits_0^{90} {{{\left( {90 – t} \right)}^{\frac{1}{6}}}dt,} \) misal \(90 – t = {u^6}{\rm{ }}\) maka \(– dt = 6{u^5} \cdot du\)
\(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\int\limits_{{{90}^{\frac{1}{6}}}}^0 { – 6{u^6}du} \)
\(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\left( {\frac{{6 \cdot {{\left( {{{90}^{\frac{1}{6}}}} \right)}^7}}}{7}} \right)\)
\(= \frac{{6 \cdot {{90}^{\frac{6}{6}}}}}{7}\)
\(= 77,142857\) |
| Jawaban | e. 77,14 |
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