Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Pemodelan dan Teori Risiko |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
20 |
SOAL
Dari sampel acak berikut ini anda membuat distribusi eksponensial:
| 1.000 |
1.400 |
5.300 |
7.400 |
7.600 |
Hitung koefisien variansi dari estimator rata-rata yang didapatkan dengan metode maximum likelihood. (Pembulatan 2 desimal)
- 0,33
- 0,45
- 0,15
- 1,00
- 1,31
| Diketahui |
Dari sampel acak berikut ini anda membuat distribusi eksponensial:
| 1.000 |
1.400 |
5.300 |
7.400 |
7.600 |
|
| Rumus yang digunakan |
\({f\left( x \right) = \frac{1}{\theta }{e^{ – \frac{x}{\theta }}},}\) \({L\left( \theta \right) = \prod\limits_{i = 1}^n {f\left( {{x_i};\theta } \right)} ,}\) \({\frac{{d\ln \left[ {L\left( \theta \right)} \right]}}{{d\theta }} = 0}\)
\({I\left( X \right)Var\left[ X \right] = 1,}\) \({CV = \frac{{\sqrt {Var\left[ X \right]} }}{{E\left[ X \right]}}}\) |
| Proses pengerjaan |
\(L\left( \theta \right) = \prod\limits_{i = 1}^n {\frac{1}{\theta }\exp \left[ { – \frac{{{x_i}}}{\theta }} \right]} = \frac{1}{{{\theta ^n}}}\exp \left[ { – \frac{{\sum\limits_{i = i}^n {{x_i}} }}{\theta }} \right]\)
\(\ln \left[ {L\left( \theta \right)} \right] = – n\ln \theta – \frac{1}{\theta }\sum\limits_{i = i}^n {{x_i}} \)
\(\frac{{d\ln \left[ {L\left( \theta \right)} \right]}}{{d\theta }} = – \frac{n}{\theta } + \frac{1}{{{\theta ^2}}}\sum\limits_{i = i}^n {{x_i}} = 0\)
\(\hat \theta = \frac{1}{n}\sum\limits_{i = i}^n {{x_i}} = \bar x\)
\(E\left[ X \right] = \hat \theta \) |
|
\(I\left( X \right) = – E\left[ {\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}}} \right] = – E\left[ {\frac{n}{{{\theta ^2}}} – \frac{2}{{{\theta ^3}}}\sum\limits_{i = i}^n {{x_i}} } \right] = – \frac{n}{{{\theta ^2}}} + \frac{{2n\theta }}{{{\theta ^3}}} = \frac{n}{{{\theta ^2}}}\)
\(Var\left[ X \right] = \frac{1}{{I\left( X \right)}} = \frac{{{{\hat \theta }^2}}}{n}\) |
|
\(CV = \frac{{\sqrt {Var\left[ X \right]} }}{{E\left[ X \right]}} = \frac{{\hat \theta }}{{\sqrt n \cdot \hat \theta }} = \frac{1}{{\sqrt 5 }} = 0.447\) |
| Jawaban |
b. 0,45 |
