Pembahasan Ujian PAI: A60 - No. 18 - November 2015

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Matematika Aktuaria
Periode Ujian	:	November 2015
Nomor Soal	:	18

SOAL

Diberikan suatu fungsi survival:

\({{S_0}\left( t \right) = 1 – {{\left( {0,01t} \right)}^2};}\) \({0 \le t \le 100}\)

Hitunglah \(e_{30:\left. {\overline {\, {50} \,}}\! \right| }^0\) (pembulatan terdekat)

Diketahui	\({{S_0}\left( t \right) = 1 – {{\left( {0,01t} \right)}^2};}\) \({0 \le t \le 100}\)
Rumus yang digunakan	\({}_t{p_x} = \frac{{{S_0}\left( {x + t} \right)}}{{{S_0}\left( x \right)}}\) \(e_{x:\left. {\overline {\, n \,}}\! \right\| }^0 = \int\limits_0^n {{}_t{p_x}dt} \)
Proses pengerjaan	\(e_{30:\left. {\overline {\, {50} \,}}\! \right\| }^0 = \int\limits_0^{50} {{}_t{p_{30}}dt} = \int\limits_0^{50} {\frac{{{S_0}\left( {30 + t} \right)}}{{{S_0}\left( {30} \right)}}dt} \) \(e_{30:\left. {\overline {\, {50} \,}}\! \right\| }^0 = \frac{{\int\limits_{30}^{80} {\left[ {1 – {{\left( {0,01t} \right)}^2}} \right]dt} }}{{1 – {{\left( {0,01\left( {30} \right)} \right)}^2}}}\) \(e_{30:\left. {\overline {\, {50} \,}}\! \right\| }^0 = \frac{1}{{0.91}}\left[ {\int\limits_{30}^{80} {1dt} – {{0.01}^2}\int\limits_{30}^{80} {{t^2}dt} } \right]\) \(e_{30:\left. {\overline {\, {50} \,}}\! \right\| }^0 = \frac{1}{{0.91}}\left[ {50 – \frac{{{{0.01}^2}\left( {{{80}^3} – {{50}^3}} \right)}}{3}} \right]\) \(e_{30:\left. {\overline {\, {50} \,}}\! \right\| }^0 = 37.179\)
Jawaban	d. 37