Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Hitunglah \(S\left( {10} \right)\) dan \(\Lambda \left( {10} \right)\) dengan metode product-limit (dibulatkan 4 desimal):
- \(S\left( {10} \right) = 0,1667;\Lambda \left( {10} \right) = 1,7918\)
- \(S\left( {10} \right) = 0,3333;\Lambda \left( {10} \right) = 1,0986\)
- \(S\left( {10} \right) = 0,2000;\Lambda \left( {10} \right) = 1,6094\)
- \(S\left( {10} \right) = 0,2771;\Lambda \left( {10} \right) = 1,2833\)
- \(S\left( {10} \right) = 0,2346;\Lambda \left( {10} \right) = 1,4500\)
| Diketahui | 6 tikus yang baru lahir. Karena kesehatan yang buruk dari induknya, tikus-tikus tersebut mati pada waktu 2; 4; 6; 9; 10; 12 |
| Rumus yang digunakan | Product-Limit \(S\left( t \right) = \prod\limits_{j = 1}^m {\left( {\frac{{{r_j} – {d_j}}}{{{r_j}}}} \right)} ,{t_m} \le t < {t_{m + 1}}\) \(\Lambda \left( t \right) = – \ln \left[ {S\left( t \right)} \right]\) |
| Proses pengerjaan | \(S\left( {10} \right) = \prod\limits_{j = 1}^5 {\left( {\frac{{{r_j} – {d_j}}}{{{r_j}}}} \right)} \) \(= \frac{5}{6} \cdot \frac{4}{5} \cdot \frac{3}{4} \cdot \frac{2}{3} \cdot \frac{1}{2}\) \(= 0,16666667\) |
| \(\Lambda \left( {10} \right) = – \ln \left[ {S\left( {10} \right)} \right]\) \(= – \ln \left[ {\frac{1}{6}} \right]\) \(= 1,791759\) | |
| Jawaban | a. \(S\left( {10} \right) = 0,1667;\Lambda \left( {10} \right) = 1,7918\) |