Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
April 2019 |
| Nomor Soal |
: |
25 |
SOAL
Untuk sebuah table double decrement, diberikan:
| Usia \(\left( x \right)\) |
\(l_x^{\left( \tau \right)}\) |
\(d_x^{\left( 1 \right)}\) |
\(d_x^{\left( 2 \right)}\) |
| 40 |
1000 |
60 |
55 |
| 41 |
– |
– |
70 |
| 42 |
750 |
– |
– |
Setiap decrement menyebar secara uniform, hitunglah nilai \(q_{41}^{‘\left( 1 \right)}\)
- 0,077
- 0,078
- 0,079
- 0,080
- 0,081
| Diketahui |
| Usia \(\left( x \right)\) |
\(l_x^{\left( \tau \right)}\) |
\(d_x^{\left( 1 \right)}\) |
\(d_x^{\left( 2 \right)}\) |
| 40 |
1000 |
60 |
55 |
| 41 |
– |
– |
70 |
| 42 |
750 |
– |
– |
|
| Rumus yang digunakan |
\({}_tp_x^{‘\left( j \right)} = 1 – {}_tq_x^{‘\left( j \right)}\), \(d_x^{\left( \tau \right)} = l_x^{\left( \tau \right)} – l_{x + 1}^{\left( \tau \right)}\), \(d_x^{\left( \tau \right)} = d_x^{\left( 1 \right)} + d_x^{\left( 2 \right)}\),
\(d_x^{\left( j \right)} = l_x^{\left( \tau \right)}q_x^{\left( j \right)}\), \(d_x^{\left( \tau \right)} = l_x^{\left( \tau \right)}q_x^{\left( \tau \right)}\),
\(p_x^{‘\left( j \right)} = {\left( {p_x^{\left( \tau \right)}} \right)^{\frac{{q_x^{\left( j \right)}}}{{q_x^{\left( \tau \right)}}}}}\) |
| Proses pengerjaan |
\(l_{41}^{\left( \tau \right)} = l_{40}^{\left( \tau \right)} – d_{40}^{\left( 1 \right)} – d_{40}^{\left( 2 \right)}\)
\(= 1000 – 60 – 55\)
\(= 885\) |
| |
\(l_{42}^{\left( \tau \right)} = l_{41}^{\left( \tau \right)} – d_{41}^{\left( 1 \right)} – d_{41}^{\left( 2 \right)}\)
\(750 = 885 – d_{41}^{\left( 1 \right)} – 70\)
\(d_{41}^{\left( 1 \right)} = 65\) |
| |
\(q_{41}^{\left( 1 \right)} = \frac{{d_{41}^{\left( 1 \right)}}}{{l_{41}^{\left( \tau \right)}}} = \frac{{65}}{{885}} = 0,073446\) dan \(q_{41}^{\left( \tau \right)} = \frac{{d_{41}^{\left( \tau \right)}}}{{l_{41}^{\left( \tau \right)}}} = \frac{{65 + 70}}{{885}} = 0,152542\)
\(q_{41}^{‘\left( 1 \right)} = 1 – p_{41}^{‘\left( 1 \right)}\)
\(= 1 – {\left( {p_{41}^{\left( \tau \right)}} \right)^{\frac{{q_{41}^{\left( 1 \right)}}}{{q_{41}^{\left( \tau \right)}}}}}\)
\(= 1 – {\left( {1 – 0,152542} \right)^{\frac{{0,073446}}{{0,152542}}}}\)
\(= 0,076599\) |
| Jawaban |
a. 0,077 |
