Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Permodelan dan Teori Risiko |
Periode Ujian |
: |
Juni 2015 |
Nomor Soal |
: |
1 |
SOAL
Sebuah distribusi Gamma memiliki rata-rata/ (“mean”) = 8 dan skewness = 1. Hitung Variansinya?
- 12
- 16
- 61
- 8
Diketahui |
- rata-rata/ ( “mean” ) = 8
- skewness = 1
- \({{\mu ‘}_k}\) : raw moment ke-k
- \(\mu k\) : central moment ke-k
|
Rumus yang digunakan |
- \(\bar X = {\mu _{10}} = \alpha \theta \)
- \(skewness = \frac{{{\mu _3}}}{\sigma }\)
- \(Varians = {\sigma ^2} = \alpha {\theta ^2}\)
|
Proses pengerjaan |
\(\mu = \bar X = {{\mu ‘}_1} = \alpha \theta \)
\({{\mu ‘}_2} = {\alpha ^2}{\theta ^2} – \alpha {\theta ^2}\)
\({{\mu ‘}_3} = {\rm{ }}({\alpha ^3} – 3{\alpha ^2} + 2\alpha ){\theta ^3}\)
\(\mu 3 = {{\mu ‘}_3} – 3{{\mu ‘}_2}\mu + 2{\mu ^3} = 2\alpha {\theta ^3}\)
\(skewness = {\mu ^3}{\sigma ^3} = \frac{{2\alpha {\theta ^3}}}{{{{({\sigma ^3}^{/2})}^2}}} = 1 \to \alpha = 4\)
Diketahui pula bahwa \(\mu = \alpha \theta = 8\) , dengan demikian kita dapatkan \(\theta = 2\)
\(Varians = {\sigma ^2} = \alpha {\theta ^2} = 4{(2)^2} = 16\) |
Jawaban |
b. 16 |