Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2016 |
| Nomor Soal |
: |
6 |
SOAL
Berdasarkan soal nomor 5, hitunglah \(Var\left[ {\left. {\hat S\left( 5 \right)} \right|\left\{ {{{n’}_i}} \right\}} \right]\)
- 0,000022
- 0,000029
- 0,000034
- 0,000041
- 0,000046
| Diketahui |
Dalam sebuah studi menggunakan pendekatan estimasi moment, diperoleh data jumlah kematian dalam interval \(\left( {x,x + 1} \right]\), berdasarkan besaran exposure yang diberikan sebagai berikut:
| Selang |
Jumlah Kematian |
Exposure |
| \(\left( {0,1} \right]\) |
12 |
1100 |
| \(\left( {1,2} \right]\) |
9 |
1220 |
| \(\left( {2,3} \right]\) |
7 |
1365 |
| \(\left( {3,4} \right]\) |
5 |
1522 |
| \(\left( {4,5} \right]\) |
4 |
1784 |
\(\hat S\left( 5 \right) = 0,971368\) |
| Rumus yang digunakan |
- \(Var\left[ {\left. {\hat S\left( 5 \right)} \right|\left\{ {{{n’}_i}} \right\}} \right] = {\left[ {\hat S\left( t \right)} \right]^2} \cdot \sum\limits_{i = 0}^{x – 1} {\left( {\frac{{{q_i}}}{{{p_i} \cdot {n_i}}}} \right)} \)
- \({p_i} = 1 – {q_i} = 1 – \frac{{{d_i}}}{{{n_i}}}\)
|
| Proses pengerjaan |
| \(i\) |
\({d_i}\) |
\({n_i}\) |
\({q_i}\) |
\({p_i}\) |
\({p_i} \cdot {n_i}\) |
\(\frac{{{q_i}}}{{{p_i} \cdot {n_i}}}\) |
| 0 |
12 |
1100 |
0.010909 |
0.989091 |
1088 |
0.00001003 |
| 1 |
9 |
1220 |
0.007377 |
0.992623 |
1211 |
0.00000609 |
| 2 |
7 |
1365 |
0.005128 |
0.994872 |
1358 |
0.00000378 |
| 3 |
5 |
1522 |
0.003285 |
0.996715 |
1517 |
0.00000217 |
| 4 |
4 |
1784 |
0.002242 |
0.997758 |
1780 |
0.00000126 |
|
0.00002332 |
\(Var\left[ {\left. {\hat S\left( 5 \right)} \right|\left\{ {{{n’}_i}} \right\}} \right] = {\left( {0.971368} \right)^2}\left[ {0.00002332} \right]\)
\(Var\left[ {\left. {\hat S\left( 5 \right)} \right|\left\{ {{{n’}_i}} \right\}} \right] = 0.000022\) |
| Jawaban |
A. 0,000022 |
