Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2016 |
| Nomor Soal |
: |
7 |
SOAL
Pada sebuah model double decrement, diperoleh informasi sebagai berikut:
\(l_x^{(\tau )} = 100\)
\(l_{x + 3}^{(\tau )} = 60\)
\(_3q_x^{(1)} = 0.04\)
\(_{2|}q_x^{(2)} = 0.06\)
Hitunglah \(q_x^{(2)}\)
- 0,3
- 0,32
- 0,35
- 0,38
- 0,4
| Diketahui |
\(l_x^{(\tau )} = 100\)
\(l_{x + 3}^{(\tau )} = 60\)
\(_3q_x^{(1)} = 0.04\)
\(_{2|}q_x^{(2)} = 0.06\) |
| Rumus yang digunakan |
\(l_{x + k}^{(\tau )} = l_x^{(\tau )} – d_x^{(1)} – d_x^{(2)}\)
\(_tq_x^{(j)} = \frac{{d_x^{(j)} + d_{x + 1}^{(j)} + … + d_{x + t – 1}^{(j)}}}{{l_x^{(\tau )}}}\)
\(_{t|u}q_x^{(j)} = \frac{{_ud_{x + t}^{(j)}}}{{l_x^{(\tau )}}}\) |
| Proses pengerjaan |
\(_3q_x^{(1)} = \frac{{d_x^{(1)} + d_{x + 1}^{(1)} + d_{x + 2}^{(1)}}}{{l_x^{(\tau )}}}\)
\(\Leftrightarrow 0.04 = \frac{{d_x^{(1)} + d_{x + 1}^{(1)} + d_{x + 2}^{(1)}}}{{100}}\)
\(\Leftrightarrow 4 = d_x^{(1)} + d_{x + 1}^{(1)} + d_{x + 2}^{(1)}{\rm{ (*)}}\)
\(_{2|}q_x^{(2)} = \frac{{d_{x + 2}^{(2)}}}{{l_x^{(\tau )}}}\)
\(\Leftrightarrow 0.06 = \frac{{d_{x + 2}^{(2)}}}{{100}}\)
\(\Leftrightarrow 6 = d_{x + 2}^{(2)}{\rm{ (**)}}\)
\(l_{x + 3}^{(\tau )} = l_x^{(\tau )} – d_x^{(1)} – d_x^{(2)} – d_{x + 1}^{(1)} – d_{x + 1}^{(2)} – d_{x + 2}^{(1)} – d_{x + 2}^{(2)}\)
\(\Leftrightarrow l_{x + 3}^{(\tau )} = l_x^{(\tau )} – (d_x^{(1)} + d_{x + 1}^{(1)} + d_{x + 2}^{(1)}) – (d_x^{(2)} + d_{x + 1}^{(2)}) – d_{x + 2}^{(2)}\)
\(\Leftrightarrow d_x^{(2)} + d_{x + 1}^{(2)} = 100 – 4 – 6 – 60\)
\(\Leftrightarrow d_x^{(2)} + d_{x + 1}^{(2)} = 30{\rm{ (***)}}\)
selanjutnya diperoleh berdasarkan ${\rm{(***)}}$
\(_2q_2^{(2)} = \frac{{d_x^{(2)} + d_{x + 1}^{(2)}}}{{l_x^{(\tau )}}} = \frac{{30}}{{100}} = 0.3\) |
| Jawaban |
a. 0,3 |
