Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | April 2019 |
| Nomor Soal | : | 28 |
SOAL
Data pembayaran klaim dari 10 polis adalah:
\(2\) \(3\) \(3\) \(5\) \({{5^ + }}\) \(6\) \(7\) \({{7^ + }}\) \(9\) \({{{10}^ + }}\)
Tanda + mengindikasikan bahwa kerugian melebihi limit polis.
Dengan menggunakan Product Limit estimator, tentukan probabilitas bahwa kerugian yang terjadi pada polis melebihi 8.
- 0,40
- 0,36
- 0,30
- 0,25
- 0,20
| Diketahui | Data pembayaran klaim dari 10 polis adalah: \(2\) \(3\) \(3\) \(5\) \({{5^ + }}\) \(6\) \(7\) \({{7^ + }}\) \(9\) \({{{10}^ + }}\) Tanda + mengindikasikan bahwa kerugian melebihi limit polis. |
| Rumus yang digunakan | \(\hat S\left( t \right) = \prod\limits_{j = 1}^m {\left( {\frac{{{r_j} – {d_j}}}{{{r_j}}}} \right)} , {t_m} \le t < {t_{m + 1}}\) |
| Proses pengerjaan | Buat table data| \(i\) | \({d_i}\) entry | \({x_i}\) event | \({u_i}\) censored | | 1 | 0 | 2 | – | | 2 | 0 | 3 | – | | 3 | 0 | 3 | – | | 4 | 0 | 5 | – | | 5 | 0 | – | 5 | | 6 | 0 | 6 | – | | 7 | 0 | 7 | – | | 8 | 0 | – | 7 | | 9 | 0 | 9 | – | | 10 | 0 | – | 10 |
Table survival | \(j\) | \({t_j}\) | \({d_j}\) | \({r_j}\) | | 1 | 2 | 1 | 10 | | 2 | 3 | 2 | 9 | | 3 | 5 | 1 | 7 | | 4 | 6 | 1 | 5 | | 5 | 7 | 1 | 4 | | 6 | 9 | 1 | 2 |
Fungsi Survival | \(t\) | \(\hat S\left( t \right)\) | | \(0 \le t < 2\) | 1 | | \(2 \le t < 3\) | \(1 – \frac{1}{{10}} = 0,9\) | | \(3 \le t < 5\) | \(\left( {0,9} \right)\left( {1 – \frac{2}{9}} \right) = 0,7\) | | \(5 \le t < 6\) | \(\left( {0,7} \right)\left( {1 – \frac{1}{7}} \right) = 0,6\) | | \(6 \le t < 7\) | \(\left( {0,6} \right)\left( {1 – \frac{1}{5}} \right) = 0,48\) | | \(7 \le t < 9\) | \(\left( {0,48} \right)\left( {1 – \frac{1}{4}} \right) = 0,36\) | | \(t \ge 9\) | \(\left( {0,36} \right)\left( {1 – \frac{1}{2}} \right) = 0,18\) |
Diperoleh \(\hat S\left( 8 \right) = \hat S\left( 7 \right) = 0,36\) |
| Jawaban | b. 0,36 |
