Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Juni 2015 |
| Nomor Soal |
: |
9 |
SOAL
Berdasarkan soal nomor 8
Hitunglah \(Var\left[ X \right]\)
- 6.075
- 5.951,02
- 5.625,00
- 247,9592
- 123,9796
| Diketahui |
\(S\left( x \right) = \frac{{{{90}^6} – {x^6}}}{{{{90}^6}}}\) atau \(0 < x < 90\) |
| Rumus yang digunakan |
\(Var\left[ X \right] = E\left[ {{X^2}} \right] – E{\left[ X \right]^2}\)
\(E\left[ {{X^2}} \right] = 2\int\limits_0^\infty {t \cdot {}_t{p_x}dt} \) |
| Proses pengerjaan |
\(E\left[ {{X^2}} \right] = 2\int\limits_0^{90} {t \cdot \frac{{{{90}^6} – {t^6}}}{{{{90}^6}}}dt} = 2\left[ {\int\limits_0^{90} {tdt} – \int\limits_0^{90} {\frac{{{t^7}}}{{{{90}^6}}}dt} } \right] = 2\left[ {\frac{{{{90}^2}}}{2} – \frac{{{{90}^8}}}{{8\left( {{{90}^6}} \right)}}} \right] = 6075\) |
|
\(Var\left[ X \right] = E\left[ {{X^2}} \right] – E{\left[ X \right]^2} = 6075 – {\left( {\frac{{540}}{7}} \right)^2} = \frac{{6075}}{{49}} = 123.979592\) |
| Jawaban |
e. 123,9796 |
