Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Permodelan dan Teori Risiko |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
16 |
SOAL
500 buah klaim kerugian asuransi kargo teramati. 5 kerugian yang teramati tersebut sebesar
| 1.100 |
3.200 |
3.300 |
3.500 |
3.900 |
Semua informasi yang diketahui mengenai 495 klaim sisanya ialah bahwa mereka masing-masing lebih dari 4.000. Tentukan parameter mean dari distribusi eksponensial menggunakan metode maximum likelihood estimate
- 933.000
- 233.000
- 333.000
- 999.000
- 399.000
| Rumus |
Exponential Distribution \((\theta )\),
- \(f(x) = \frac{{{e^{ – \frac{x}{\theta }}}}}{\theta }\)
- \(S(x) = {e^{ – \frac{x}{\theta }}}\)
|
| Step 1 |
\(L = f(1.100)f(3.200)f(3.300)f(3.500)f(3.900){\left[ {S(4.000)} \right]^{495}}\)
\(L = \left( {\frac{{{e^{ – \frac{{1.100}}{\theta }}}}}{\theta }} \right)\left( {\frac{{{e^{ – \frac{{3.200}}{\theta }}}}}{\theta }} \right)\left( {\frac{{{e^{ – \frac{{3.300}}{\theta }}}}}{\theta }} \right)\left( {\frac{{{e^{ – \frac{{3.500}}{\theta }}}}}{\theta }} \right)\left( {\frac{{{e^{ – \frac{{3.900}}{\theta }}}}}{\theta }} \right){\left[ {{e^{ – \frac{{4.000}}{\theta }}}} \right]^{495}}\)
\(L = {\theta ^{ – 5}}\left( {{e^{ – \frac{{(1.100 + 3.200 + 3.300 + 3.500 + 3.900)}}{\theta }}}} \right){\left[ {{e^{ – \frac{{4.000}}{\theta }}}} \right]^{495}}\)
\(L = {\theta ^{ – 5}}\left( {{e^{ – \frac{{15.000}}{\theta }}}} \right)\left[ {{e^{ – \frac{{1.980.000}}{\theta }}}} \right]\)
\(L = {\theta ^{ – 5}}\left( {{e^{ – \frac{{1.995.000}}{\theta }}}} \right)\) |
| Step 2 |
- \(\frac{{d\ln (L)}}{{d\theta }} = 0\)
\(\ln (L) = \ln ({\theta ^{ – 5}}{e^{ – \frac{{1.995.000}}{\theta }}})\)
\(\ln (L) = \ln ({\theta ^{ – 5}}) + ln({e^{ – \frac{{1.995.000}}{\theta }}})\)
\(\ln (L) = – 5\ln (\theta ) – \frac{{1.995.000}}{\theta }\)
\(\frac{{d\ln (L)}}{{d\theta }} = \frac{{ – 5}}{\theta } + \frac{{1.995.000}}{{{\theta ^2}}}\)
\(0 = \frac{{ – 5}}{\theta } + \frac{{1.995.000}}{{{\theta ^2}}}\)
\(5\theta = 1.995.000\)
\(\hat \theta = 399.000\) |
| Jawaban |
e. 399.000 |
