Pembahasan Ujian PAI: A50 - No. 12 - Agustus 2023

29 July, 2026

Pembahasan Ujian PAI: A50 – No. 12 – Agustus 2023

7 July 2026

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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