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 |