Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Metoda Statistika |
Periode Ujian |
: |
November 2014 |
Nomor Soal |
: |
24 |
SOAL
Anda diberi informasi berikut tentang suatu model data berkala statis (time series stationer):
\({\rho _1} = – 0,310\)
\({\rho _2} = – 0,155\)
\({\rho _k} = 0;k = 3,4,5, \ldots \)
Selain itu, anda juga diberikan informasi:
\({\theta _1} + {\theta _2} = 0,7\)
Berpakah nilai \({\theta _1}\)
- 0,2
- 0,3
- 0,4
- 0,5
- 0,6
Diketahui |
Suatu model data berkala statis (time series stationer) memiliki:
- \({\rho _1} = – 0,310\)
\({\rho _2} = – 0,155\)
\({\rho _k} = 0;k = 3,4,5, \ldots \)
- \({\theta _1} + {\theta _2} = 0,7\)
|
Rumus yang digunakan |
Karena ada \({\theta _1}\) dan \({\theta _2}\) maka model time series-nya adalah \(MA\left( 2 \right)\) dengan fungsi autokorelasi:
\({\rho _1} = \frac{{ – {\theta _1}\left( {1 – {\theta _2}} \right)}}{{1 + \theta _1^2 + \theta _2^2}}\) dan \({\rho _2} = \frac{{ – {\theta _2}}}{{1 + \theta _1^2 + \theta _2^2}}\) |
Proses pengerjaan |
- \({\rho _2} = \frac{{ – {\theta _2}}}{{1 + \theta _1^2 + \theta _2^2}}\)
\(– 0.155 = \frac{{ – \left( {0.7 – {\theta _1}} \right)}}{{1 + \theta _1^2 + {{\left( {0.7 – {\theta _1}} \right)}^2}}}\)
\(0.155 + 0.155\theta _1^2 + 0.07595 – 0.217{\theta _1} + 0.155\theta _1^2 = 0.7 – {\theta _1}\)
\(0.31\theta _1^2 + 0.783{\theta _1} – 0.46905 = 0\)
- dengan rumus abc:
\({\theta _1} = \frac{{ – 0.783 \pm \sqrt {{{0.783}^2} – 4\left( {0.31} \right)\left( { – 0.46905} \right)} }}{{2\left( {0.31} \right)}}\)
\({\theta _1} = \frac{{ – 0.783 \pm 1.093028}}{{0.62}}\)
\({\theta _1} = 0.500045\) atau ${\theta _1} = – 3.025852$
|
Jawaban |
D. 0,5 |