Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Pemodelan dan Teori Risiko |
| Periode Ujian | : | November 2018 |
| Nomor Soal | : | 18 |
SOAL
Diberikan
| Banyaknya Klaim | Peluang | Besar Klaim | Peluang |
| 0 | \(\frac{1}{5}\) | ||
| 1 | \(\frac{3}{5}\) | 25 | \(\frac{1}{3}\) |
| 150 | \(\frac{2}{3}\) | ||
| 2 | \(\frac{1}{5}\) | 50 | \(\frac{2}{3}\) |
| 200 | \(\frac{1}{3}\) |
Masing-masing besar klaim ialah saling bebas. Hitung variansi dari kerugian gabungan (aggregate loss)
- 4.050
- 8.100
- 10.500
- 12.100
- 15.930
| Diketahui |
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| Rumus yang digunakan | Bernoulli: \(Var\left[ S \right] = {\left( {b – a} \right)^2}q\left( {1 – q} \right)\) \({E\left[ S \right] = E\left[ N \right]E\left[ X \right],}\) \({\mu = \sum\limits_{i = 1}^n {n \cdot \Pr \left( S \right)} ,}\) \({{\sigma ^2} = \sum\limits_{i = 1}^n {{n^2} \cdot \Pr \left( S \right)} – {\mu ^2}}\) | ||||||||||||||||||||
| Proses pengerjaan | Varians dari aggregate klaim berdasarkan besar klaim (menggunakan distribusi Bernoulli karena hnaya ada 2 peluang)
Diperoleh nilai ekspektasi dari varians \(E\left[ {Var\left( {\left. S \right|N} \right)} \right] = \left( {\frac{3}{5}} \right)\left( {\frac{{2\left( {{{125}^2}} \right)}}{9}} \right) + \left( {\frac{1}{5}} \right)\left( {10,000} \right) = \frac{{12250}}{3} = 4,083\frac{1}{3}\) | ||||||||||||||||||||
Rata-rata dari aggregate klaim
Diperoleh nilai varians dari mean \(Var\left[ {E\left( {\left. S \right|N} \right)} \right] = \left[ {\left( {\frac{3}{5}} \right){{\left( {\frac{{325}}{3}} \right)}^2} + \left( {\frac{1}{5}} \right){{\left( {200} \right)}^2}} \right] – {\left[ {\left( {\frac{3}{5}} \right)\left( {\frac{{325}}{3}} \right) + \left( {\frac{1}{5}} \right)\left( {200} \right)} \right]^2}\) \(Var\left[ {E\left( {\left. S \right|N} \right)} \right] = \frac{{45,125}}{3} – {105^2} = \frac{{12,050}}{3} = 4,016\frac{2}{3}\) | |||||||||||||||||||||
| Jadi varians dari aggregate loss \(Var\left[ S \right] = E\left[ {Var\left( {\left. S \right|N} \right)} \right] + Var\left[ {E\left( {\left. S \right|N} \right)} \right] = 4,083\frac{1}{3} + 4,016\frac{2}{3} = 8,100\) | |||||||||||||||||||||
| Jawaban | b. 8.100 |