Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
A60 – Matematika Aktuaria |
| Periode Ujian |
: |
November 2017 |
| Nomor Soal |
: |
14 |
SOAL
Jika diketahui \(\mu _{x + t}^{(1)} = 0,1\) dan \(\mu _{x + t}^{(2)} = 0,2\) . Hitunglah nilai dari \({}_\infty q_x^{(1)}\)
- 1
- 1/2
- 1/3
- 1/4
- 1/5
PEMBAHASAN
| Rumus |
\({}_\infty q_x^{(1)} = \int\limits_0^\infty {_tP_x^{(\tau )}\,\mu _{x + t}^{(1)}\,\,dt} \)
\(\mu _{x + t}^{(\tau )} = \mu _{x + t}^{(1)} + \mu _{x + t}^{(2)}\)
\(_tP_x^{(\tau )} = \exp \left[ { – \int\limits_0^t {\mu _{x + s}^{(\tau )}\,ds} } \right]\) |
| Step 1 |
\(\mu _{x + t}^{(\tau )} = 0,1 + 0,2\)
\(\mu _{x + t}^{(\tau )} = 0,3\) |
| \(_tP_x^{(\tau )} = \exp \left[ { – \int\limits_0^t {0,3\,ds} } \right]\)
\(_tP_x^{(\tau )} = \exp \left[ { – 0,3\,(t – 0)} \right]\)
\(_tP_x^{(\tau )} = \exp \left[ { – 0,3\,t} \right]\) |
| Step 2 |
\({}_\infty q_x^{(1)} = \int\limits_0^\infty {{e^{ – 0,3t}}\,0,1\,\,dt} \)
\({}_\infty q_x^{(1)} = 0,1\int\limits_0^\infty {{e^{ – 0,3t}}\,\,\,dt} \)
\({}_\infty q_x^{(1)} = \frac{{0,1}}{{ – 0,3}}{e^{ – 0,3t}}\left| {_0^\infty } \right.\)
\({}_\infty q_x^{(1)} = \frac{{0,1}}{{ – 0,3}}(0 – 1)\)
\({}_\infty q_x^{(1)} = \frac{1}{3}\) |
| Jawaban |
c. 1/3 |
