Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Juni 2016 |
| Nomor Soal |
: |
11 |
SOAL
Diketahui tiga hasil pengamatan sebagai berikut:
0,70 0,82 0,92
Anda mencocokkan sebuah distribusi dengan fungsi kepadatan (density function) berikut ini terhadap data:
\(f\left( x \right) = \left( {p + 1} \right){x^p},\) \(0 < x < 1,p > – 1\)
Hitunglah estimasi maximum likelihood atas \(p\) (dibulatkan 2 desimal).
- 2,12
- 2,67
- 3,70
- 4,32
- 6,81
| Diketahui |
tiga hasil pengamatan sebagai berikut: 0,70; 0,82; 0,92
\(f\left( x \right) = \left( {p + 1} \right){x^p},\) \(0 < x < 1,p > – 1\) |
| Rumus yang digunakan |
\(L\left( p \right) = \prod\limits_{i = 1}^n {f\left( {{x_i}} \right)} \) |
| Proses pengerjaan |
\(L\left( p \right) = \prod\limits_{i = 1}^n {f\left( {{x_i}} \right)} \)
\(= \prod\limits_{i = 1}^n {\left( {p + 1} \right){x_i}^p} \)
\(= {\left( {p + 1} \right)^n} \cdot {\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)^p}\)
\(\ln \left[ {L\left( p \right)} \right] = n \cdot \ln \left( {p + 1} \right) + p \cdot \ln \left( {\prod\limits_{i = 1}^n {{x_i}} } \right)\)
\(= n \cdot \ln \left( {p + 1} \right) + p \cdot \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} \)
\(\frac{{d\ln \left[ {L\left( p \right)} \right]}}{{dp}} = \frac{n}{{\hat p + 1}} + \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} = 0\)
\(– n = \hat p \cdot \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} + \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} \)
\(\hat p = \frac{{ – n – \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} }}{{\sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} }}\) |
|
Sehingga,
\(\hat p = \frac{{ – n – \sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} }}{{\sum\limits_{i = 1}^n {\ln \left( {{x_i}} \right)} }}\)
\(= \frac{{ – 3 – \left( {\ln \left( {0,7} \right) + \ln \left( {0,82} \right) + \ln \left( {0,92} \right)} \right)}}{{\ln \left( {0,7} \right) + \ln \left( {0,82} \right) + \ln \left( {0,92} \right)}}\)
\(= 3,698457\) |
| Jawaban |
c. 3,70 |
