Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | April 2019 |
| Nomor Soal | : | 22 |
SOAL
Diketahui
| Waktu (t) | Jumlah yang beresiko saat t | Jumlah kegagalan saat t |
| 1 | 30 | 5 |
| 2 | 27 | 9 |
| 3 | 32 | 6 |
| 4 | 25 | 5 |
| 5 | 20 | 4 |
Tentukan aproksimasi Greenwood’s dari variansi \({}_3{\hat p_1}\)
- 0,0067
- 0,0073
- 0,0080
- 0,0091
- 0,0105
| Diketahui | | Waktu (t) | Jumlah yang beresiko saat t | Jumlah kegagalan saat t | | 1 | 30 | 5 | | 2 | 27 | 9 | | 3 | 32 | 6 | | 4 | 25 | 5 | | 5 | 20 | 4 |
|
| Rumus yang digunakan | \(\widehat {Var}\left[ {{}_t{{\hat p}_x}} \right] = {\left[ {{}_t{{\hat p}_x}} \right]^2} \cdot \sum\limits_{j = 1}^k {\left( {\frac{{{d_j}}}{{{r_j}\left( {{r_j} – {d_j}} \right)}}} \right)} \)
\({}_t{\hat p_x} = \hat S\left( {{t_k}} \right) = \prod\limits_{j = 1}^k {\left( {\frac{{{r_j} – {d_j}}}{{{r_j}}}} \right)} {\rm{ }}\) untuk \({t_k} \le t < {t_{k + 1}}\) |
| Proses pengerjaan | \({}_3{\hat p_1} = \left( {\frac{{27 – 9}}{{27}}} \right)\left( {\frac{{32 – 6}}{{32}}} \right)\left( {\frac{{25 – 5}}{{25}}} \right) = 0,433333\) |
| \(\widehat {Var}\left[ {{}_3{{\hat p}_1}} \right] = {\left[ {{}_3{{\hat p}_1}} \right]^2} \cdot \sum\limits_{j = 2}^4 {\left( {\frac{{{d_j}}}{{{r_j}\left( {{r_j} – {d_j}} \right)}}} \right)} \)
\(= {\left( {0,433333} \right)^2} \cdot \left( {\frac{9}{{\left( {27} \right)\left( {18} \right)}} + \frac{6}{{\left( {32} \right)\left( {26} \right)}} + \frac{5}{{\left( {25} \right)\left( {20} \right)}}} \right)\)
\(= 0,006709\) |
| Jawaban | a. 0,0067 |
