Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Probabilita dan Statistika |
| Periode Ujian |
: |
Juni 2014 |
| Nomor Soal |
: |
27 |
SOAL
Seorang ahli statistik membuat pemodelan dari jumlah gol per 90 menit pertandingan sepakbola antara klub MU melawan MC. Dari model yang diperoleh, jumlah gol per 90 menit pertandingan yang dibuat oleh MU memiliki distribusi Geometri, X = 0, 1, 2, … dengan mean 3,5. Sedangkan jumlah gol per 90 menit pertandingan yang dibuat oleh MC juga berdistribusi Geometri, Y = 0, 1, 2, … dengan mean 3,0. Jika diasumsikan X dan Y saling bebas, maka probabilitas MC menang dengan selisih 2 gol atau lebih, dalam 90 menit pertandingan melawan MU sama dengan …
- 10%
- 20%
- 30%
- 40%
- 50%
| Diketahui |
\({\mu _X} = \frac{{1 – P}}{P} = 3,5\)
\({\mu _Y} = \frac{{1 – P}}{P} = 3,0\) |
| Rumus yang digunakan |
\(\Pr (Y – X \ge 2) = \Pr \left( {Z \ge \frac{{(Y – X) – \left( {{\mu _Y} – {\mu _X}} \right)}}{{\sqrt {\sigma _Y^2 + \sigma _X^2} }}} \right)\) |
| Proses pengerjaan |
\({\mu _X} = \frac{{1 – P}}{P} = 3,5\)
sehingga
\(\sigma _X^2 = \frac{{1 – P}}{{{P^2}}} = \frac{{63}}{4}\)
sedangkan dengan
\({\mu _Y} = \frac{{1 – P}}{P} = 3,0\) diperoleh:
\(\sigma _Y^2 = \frac{{1 – P}}{{{P^2}}} = 12\)
\(\Pr (Y – X \ge 2) = \Pr \left( {\frac{{(Y – X) – \left( {{\mu _Y} – {\mu _X}} \right)}}{{\sqrt {\sigma _Y^2 + \sigma _X^2} }} \ge \frac{{2 – \left( { – 0,5} \right)}}{{\sqrt {12 + \frac{{63}}{4}} }}} \right)\)
\(\Pr (Y – X \ge 2) = \Pr \left( {Z \ge \frac{{2 – \left( { – 0,5} \right)}}{{\sqrt {12 + \frac{{63}}{4}} }}} \right) = \Pr \left( {Z \ge \frac{{2,5}}{{\sqrt {12 + \frac{{63}}{4}} }}} \right)\)
\(\Pr (Y – X \ge 2) = \Pr \left( {Z \ge 0,4746} \right)\)
\(\Pr (Y – X \ge 2) = 1 – 0,6808 = 0,3192\) |
| Jawaban |
c. 30% |
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