Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Banyaknya klaim dapat dimodelkan melalui distribusi negatif binomial dengan parameter \(r = 15\) dan \(\beta = 15\). Jumlah besarnya klaim tersebut diketahui berdistribusi seragam/uniform pada interval \(\left( {0,10} \right)\). Menggunakan pendekatan normal, tentukan premi sehingga peluang dimana klaim melebihi premium ialah 0,05.
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| Diketahui |
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| Rumus yang digunakan | \(\Pr \left( S \right) = \frac{{s – E\left( S \right)}}{{\sqrt {Var\left( S \right)} }}\) atau \(s = E\left( S \right) + \Pr \left( S \right)\sqrt {Var\left( S \right)} \)
Negatif Binomial: \(E\left[ N \right] = r\beta \) dan \(Var\left[ N \right] = r\beta \left( {1 + \beta } \right)\)
Uniform: \(E\left[ X \right] = \frac{{b + a}}{2}\) dan \(Var\left[ X \right] = \frac{{{{\left( {b – a} \right)}^2}}}{{12}}\) Agregat: \(E\left[ S \right] = E\left[ N \right]E\left[ X \right]\) dan \(Var\left[ S \right] = E\left[ N \right]Var\left[ X \right] + Var\left[ N \right]E{\left[ X \right]^2}\) |
| Proses pengerjaan | Banyaknya Klaim Berdistribusi Negatif Binomial \(\left( {15,15} \right)\) \(E\left[ N \right] = r\beta = \left( {15} \right)\left( {15} \right) = 225\) \(Var\left[ N \right] = r\beta \left( {1 + \beta } \right) = \left( {15} \right)\left( {15} \right)\left( {1 + 15} \right) = 3600\) |
| Besarnya Klaim Berdistribusi Uniform \(\left( {0,10} \right)\) \(E\left[ X \right] = \frac{{b + a}}{2} = \frac{{10 + 0}}{2} = 5\) \(Var\left[ X \right] = \frac{{{{\left( {b – a} \right)}^2}}}{{12}} = \frac{{{{\left( {10 – 0} \right)}^2}}}{{12}} = 8.33\) | |
| Agregat Klaim \(E\left[ S \right] = E\left[ N \right]E\left[ X \right] = \left( {225} \right)\left( 5 \right) = 1125\) \(Var\left[ S \right] = E\left[ N \right]Var\left[ X \right] + Var\left[ N \right]E{\left[ X \right]^2} = \left( {225} \right)\left( {8.33} \right) + \left( {3600} \right)\left( {{5^2}} \right) = 91875\) | |
| Premi \(s = E\left( S \right) + \Pr \left( S \right)\sqrt {Var\left( S \right)} \) \(s = 1125 + 1.645\sqrt {91875} = 1623.614126\) | |
| Jawaban | 1624 (Tetapi dari PAI jawabannya B?) |