Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Banyaknya klaim pada suatu polis mempunyai distribusi Poisson dengan rata-rata \(P\). \(P\) bervariasi untuk setiap pemegang polis. \(P\) diketahui berdistribusi uniform pada selang \(\left[ {1,2} \right]\). Hitung variansi banyaknya
- \(\frac{3}{2}\)
- \(\frac{{19}}{{12}}\)
- \(\frac{5}{3}\)
- \(\frac{7}{4}\)
- \(\frac{{23}}{{12}}\)
| Diketahui |
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| Rumus yang digunakan | \(Var\left[ X \right] = E\left[ {{X^2}} \right] – {\left( {E\left[ {{X^2}} \right]} \right)^2} = E\left[ {Var\left( {\left. X \right|Y} \right)} \right] + Var\left( {E\left[ {\left. X \right|Y} \right]} \right)\) \(= E\left[ {{Y^2}} \right] + Var\left[ Y \right]\) Poisson: \(E\left[ X \right] = \lambda \) dan \(Var\left( X \right) = \lambda \) Uniform: \(E\left[ X \right] = \frac{{a + b}}{2}\) dan \(Var\left( X \right) = \frac{{{{\left( {b – a} \right)}^2}}}{{12}}\) |
| Proses pengerjaan | Sesuai distribusi poisson diperoleh \(E\left[ {\left. N \right|P} \right] = \lambda = P\) dan \(Var\left( {\left. N \right|P} \right) = \lambda = P\) |
| \(Var\left( N \right) = Var\left( {E\left[ {\left. N \right|P} \right]} \right) + E\left[ {Var\left( {\left. N \right|P} \right)} \right]\) \(Var\left( N \right) = Var\left( P \right) + E\left[ P \right]\) \(Var\left( N \right) = \frac{{{{\left( {b – a} \right)}^2}}}{{12}} + \frac{{a + b}}{2}\) \(Var\left( N \right) = \frac{1}{{12}} + \frac{3}{2} = \frac{{19}}{{12}}\) | |
| Jawaban | b. \(\frac{{19}}{{12}}\) |