Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Besaran sebuah kerugian memiliki cumulative distribution function sebagai berikut:
\(F(x){\rm{ }} = \frac{3}{4}{\left( {\frac{x}{{100}}} \right)^{\frac{1}{4}}} + \frac{1}{4}{\left( {\frac{x}{{100}}} \right)^{\frac{1}{2}}};0 \le x \le 100\)Hitunglah Loss Elimination Ratio untuk sebuah ordinary deductible sebesar 20. Pilih pembulatan terdekat.
- 0,14
- 0,20
- 0,36
- 0,42
- 0,45
| Diketahui | \(F(x){\rm{ }} = \frac{3}{4}{\left( {\frac{x}{{100}}} \right)^{\frac{1}{4}}} + \frac{1}{4}{\left( {\frac{x}{{100}}} \right)^{\frac{1}{2}}};0 \le x \le 100\) |
| Rumus yang digunakan | Loss elimination ratio = \(\frac{{E(X \wedge d)}}{{E(X)}}\) |
| Proses pengerjaan | \(E(X \wedge d){\rm{ }} = \int\limits_0^d {1 – F(x)dx = } \int\limits_0^d {\left( {\frac{3}{4}{{\left( {\frac{x}{{100}}} \right)}^{\frac{1}{4}}} + \frac{1}{4}{{\left( {\frac{x}{{100}}} \right)}^{\frac{1}{2}}}} \right)dx} \) \(E(X \wedge d){\rm{ }} = d – \frac{3}{{50}}\sqrt {10} {\left( d \right)^{\frac{5}{4}}} – \frac{1}{{60}}{\left( d \right)^{\frac{3}{2}}}\) \(E(X \wedge 20){\rm{ }} = 20 – \frac{3}{{50}}\sqrt {10} {\left( {20} \right)^{\frac{5}{4}}} – \frac{1}{{60}}{\left( {20} \right)^{\frac{3}{2}}} = 10,4844\) \(E(X) = E(X \wedge 100){\rm{ }} = 100 – \frac{3}{{50}}\sqrt {10} {\left( {100} \right)^{\frac{5}{4}}} – \frac{1}{{60}}{\left( {100} \right)^{\frac{3}{2}}} = 23,333\) Loss elimination ratio = \(\frac{{E(X \wedge 20)}}{{E(X)}} = \frac{{10,4844}}{{23,333}} = 0,45\) |
| Jawaban | E. 0,45 |