Pembahasan Soal Ujian Profesi Aktuaris
SOAL
\(X\) dan \(Y\) adalah dua peubah acak yang berdistribusi eksponensial. Diberikan \(Var\left( X \right) = 25\) dan \(Var\left( {XY} \right) = 7.500\). Tentukan \(Var\left( Y \right)\)- 25
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| Diketahui |
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| Rumus yang digunakan | Distribusi Eksponensial: \(E\left[ X \right] = \mu = \theta \) dan \(Var\left( X \right) = {\sigma ^2} = {\theta ^2}\) \(Var\left( {XY} \right) = E\left[ {{X^2}{Y^2}} \right] – {\left[ {E\left( {XY} \right)} \right]^2} = \sigma _X^2\sigma _Y^2 + \mu _Y^2\sigma _X^2 + \mu _X^2\sigma _Y^2\) |
| Proses pengerjaan | \(Var\left( X \right) = \theta _X^2 = 25\) \({\theta _X} = {\mu _X} = 5\) |
| \(Var\left( {XY} \right) = \sigma _X^2\sigma _Y^2 + \mu _Y^2\sigma _X^2 + \mu _X^2\sigma _Y^2\) \(7,500 = 25\theta _Y^2 + \left( {\theta _Y^2} \right)\left( {25} \right) + \left( {{5^2}} \right)\left( {\theta _Y^2} \right)\) \(7,500 = 25\theta _Y^2 + 25\theta _Y^2 + 25\theta _Y^2\) \(\theta _Y^2 = \frac{{7,500}}{{75}}\) \(Var\left( Y \right) = 100\) | |
| Jawaban | c. 100 |