Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Agustus 2023 |
| Nomor Soal | : | 12 |
SOAL
Diberikan informasi terkait model runtun waktu statis (stationary time series) sebagai berikut:
\(\rho_1 = -0{,}310\)
\(\rho_2 = -0{,}155\)
\(\rho_k = 0, \quad k = 3, 4, 5, \ldots\)
\(\theta_1 + \theta_2 = 0{,}7\)
Tentukan nilai \(\theta_1\)
a. 0,1
b. 0,2
c. 0,3
d. 0,4
e. 0,5
| Diketahui | \(\rho_1 = -0{,}310\),
\(\rho_2 = -0{,}155\),
\(\rho_k = 0, \quad k = 3, 4, 5, \ldots\),
\(\theta_1 + \theta_2 = 0{,}7\) |
| Rumus yang digunakan | Formula,- \(\rho_1 = \dfrac{-\theta_1 + \theta_1 \theta_2}{1 + \theta_1^2 + \theta_2^2}\)
- \(\rho_2 = \dfrac{-\theta_2}{1 + \theta_1^2 + \theta_2^2}\)
|
| Proses pengerjaan | - \(\theta_2 = 0{,}7 – \theta_1\)
- \(\dfrac{\rho_1}{\rho_2} = \dfrac{-\theta_1 + \theta_1 \theta_2}{-\theta_2}\)
\(\dfrac{-0{,}310}{-0{,}155} = \dfrac{-\theta_1 + \theta_1 \theta_2}{-\theta_2}\)
\(2\theta_2 = \theta_1 – \theta_1 \theta_2\)Substitusi \(\theta_2 = 0{,}7 – \theta_1\):
\(2(0{,}7 – \theta_1) = \theta_1 – \theta_1 (0{,}7 – \theta_1)\)
\(\theta_1^2 + 2{,}3\theta_1 – 1{,}4 = 0\)
\(\mathbf{\theta_1 = 0{,}5} \; ; \; \theta_1 = -2{,}8\) |
| Jawaban | e. 0,5 |
