492 Share 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 Kunci Jawaban & Pembahasan 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 A60AktuariaEdukasiMatematika AktuariaPAIUjian Profesi 492 Share