Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | A60 – Matematika Aktuaria |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 1 |
SOAL
Diberikan sebagai berikut :
\({S_X}(x) = \frac{{9000 – 10x – {x^2}}}{{9000}},\,\,untuk\,0 < x \le 90\)
Hitunglah nilai dari \({q_{50}} – {\mu _{50}}\)
- 0,000167
- 0,000200
- 0,000250
- 0,000333
- 0,000500
PEMBAHASAN
| Diketahui | \({S_X}(x) = \frac{{9000 – 10x – {x^2}}}{{9000}},\,\,untuk\,0 < x \le 90\) |
| Step 1 | \({\mu _X} = \frac{{ – \frac{d}{{dx}}{S_X}(x)}}{{{S_X}(x)}}\)
\({\mu _X} = \frac{{ – \frac{d}{{dx}}\frac{{9000 – 10x – {x^2}}}{{9000}}}}{{\frac{{9000 – 10x – {x^2}}}{{9000}}}}\)
\({\mu _X} = \frac{{\frac{{10 + 2x}}{{9000}}}}{{\frac{{9000 – 10x – {x^2}}}{{9000}}}}\)
\({\mu _X} = \frac{{10 + 2x}}{{9000 – 10x – {x^2}}}\) |
| \({\mu _{50}} = \frac{{10 + 2(50)}}{{9000 – 10(50) – {{(50)}^2}}}\)
\({\mu _{50}} = \frac{{110}}{{6000}}\) |
| Step 2 | \({q_{50}} = 1 – {p_{50}}\)
\({q_{50}} = 1 – \frac{{{S_X}(51)}}{{{S_X}(50)}}\)
\({q_{50}} = 1 – \frac{{\frac{{9000 – 10(51) – {{(51)}^2}}}{{9000}}}}{{\frac{{9000 – 10(50) – {{(50)}^2}}}{{9000}}}}\)
\({q_{50}} = 1 – \frac{{1963}}{{2000}}\)
\({q_{50}} = \frac{{37}}{{2000}}\) |
| Step 3 | \({q_{50}} – {\mu _{50}} = \frac{{37}}{{2000}} – \frac{{110}}{{6000}}\)
\({q_{50}} – {\mu _{50}} = \frac{1}{{6000}}\)
\({q_{50}} – {\mu _{50}} \cong 0,000167\) |
| Jawaban | a. 0,000167 |
