Pembahasan Ujian PAI: A50 – No. 14 – Juni 2016

By Fery Widhiatmoko On Mar 29, 2019 Last updated Mar 29, 2019

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Metoda Statistika
Periode Ujian	:	Juni 2016
Nomor Soal	:	14

SOAL

Diketahui

\(F\left( X \right) = 1 – {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\) untuk \(0 \le x \le 120,\)

Hitunglah \(E\left[ T \right]\) untuk \(x = 30\), yaitu \(\mathop {{e_{30}}}\limits^ \circ \)

Fery Widhiatmoko 564 posts 11 comments

Merupakan lulusan Magister Matematika minat Aktuaria dari Universitas Gadjah Mada.

Diketahui	\(F\left( X \right) = 1 – {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\) untuk \(0 \le x \le 120,\)
Rumus yang digunakan	\(S\left( X \right) = 1 – F\left( X \right)\) \({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\) \(\mathop {{e_x}}\limits^ \circ = \int\limits_0^\infty {{}_t{p_x}dt} \)
Proses Pengerjaan	\(S\left( X \right) = 1 – F\left( X \right)\) \(= {\left( {1 – \frac{x}{{120}}} \right)^{\frac{1}{6}}}\)
	\({}_t{p_x} = \frac{{S\left( {x + t} \right)}}{{S\left( x \right)}}\) \(= \frac{{{{\left( {\frac{{120 – x – t}}{{120}}} \right)}^{\frac{1}{6}}}}}{{{{\left( {\frac{{120 – x}}{{120}}} \right)}^{\frac{1}{6}}}}}\) \(= {\left( {\frac{{120 – x – t}}{{120 – x}}} \right)^{\frac{1}{6}}}\)
	\(\mathop {{e_{30}}}\limits^ \circ = \int\limits_0^{90} {{}_t{p_{30}}dt} \) \(= \int\limits_0^{90} {{{\left( {\frac{{90 – t}}{{90}}} \right)}^{\frac{1}{6}}}dt} \) \(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\int\limits_0^{90} {{{\left( {90 – t} \right)}^{\frac{1}{6}}}dt,} \) misal \(90 – t = {u^6}{\rm{ }}\) maka \(– dt = 6{u^5} \cdot du\) \(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\int\limits_{{{90}^{\frac{1}{6}}}}^0 { – 6{u^6}du} \) \(= {\left( {\frac{1}{{90}}} \right)^{\frac{1}{6}}}\left( {\frac{{6 \cdot {{\left( {{{90}^{\frac{1}{6}}}} \right)}^7}}}{7}} \right)\) \(= \frac{{6 \cdot {{90}^{\frac{6}{6}}}}}{7}\) \(= 77,142857\)
Jawaban	e. 77,14