Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Permodelan dan Teori Risiko |
| Periode Ujian | : | November 2016 |
| Nomor Soal | : | 3 |
SOAL
Kerugian dari sebuah pertanggungan asuransi berdistribusi dengan fungsi kepadatan (density function) sebagai berikut:
\(f\left( x \right) = \frac{3}{{{{100}^3}}}{\left( {100 – x} \right)^2}\) ; \(0 \le x \le 100\)
Kerugian memiliki ordinary deductible sebesar 15.
Hitunglah Loss Elimination Ratio (gunakan pembulatan terdekat!)
- 0,46
- 0,48
- 0,50
- 0,52
- 0,54
| Diketahui | Kerugian dari sebuah pertanggungan asuransi berdistribusi dengan fungsi kepadatan (density function) sebagai berikut: \(f\left( x \right) = \frac{3}{{{{100}^3}}}{\left( {100 – x} \right)^2}\) ; \(0 \le x \le 100\)
Kerugian memiliki ordinary deductible sebesar 15. |
| Rumus yang digunakan | - \(LER = \frac{{E\left[ {X \wedge u} \right]}}{{E\left[ X \right]}}\)
- \(E\left[ {X \wedge d} \right] = \int\limits_0^d {S\left( x \right)dx} \)
- \(E\left[ X \right] = E\left[ {X \wedge u} \right]\)
- \(S\left( x \right) = \int\limits_x^\infty {f\left( t \right)dt} \)
|
| Proses pengerjaan | \(S\left( x \right) = \int\limits_x^{100} {f\left( t \right)dt} = \int\limits_x^{100} {\frac{3}{{{{100}^3}}}{{\left( {100 – t} \right)}^2}dt} \)
\(S\left( x \right) = \frac{1}{{{{100}^3}}}{\left( {100 – x} \right)^3}\)
\(E\left[ {X \wedge d} \right] = \int\limits_0^d {S\left( x \right)dx} = \int\limits_0^d {\frac{1}{{{{100}^3}}}{{\left( {100 – x} \right)}^3}dx} \)
\(E\left[ {X \wedge d} \right] = \frac{1}{{\left( 4 \right){{100}^3}}}\left( {{{100}^4} – {{\left( {100 – d} \right)}^4}} \right)\)
\(E\left[ {X \wedge 15} \right] = \frac{1}{{\left( 4 \right){{100}^3}}}\left( {{{100}^4} – {{\left( {100 – 15} \right)}^4}} \right) = 11.95\)
\(E\left[ X \right] = E\left[ {X \wedge 100} \right] = \frac{1}{{4{{\left( {100} \right)}^3}}}\left( {{{100}^4}} \right) = 25\)
\(LER = \frac{{E\left[ {X \wedge 15} \right]}}{{E\left[ X \right]}} = \frac{{11.95}}{{25}} = 0.478\) |
| Jawaban | B. 0,48 |
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