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 |
