Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
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Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
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Matematika Aktuaria |
Periode Ujian |
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November 2015 |
Nomor Soal |
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23 |
SOAL
Misalkan \({X_1},{X_2}, \ldots ,{X_n}\) suatu variabel acak yang bebas, sehingga setiap \({X_i}\) memiliki “expected value \(\mu \)” dan variansi \({\sigma ^2}\). Jika \({S_n} = {X_1} + {X_2} + \cdots + {X_n}\) maka \(E\left[ {{S_n}} \right]\) adalah
- \(\mu \)
- \(\frac{\mu }{n}\)
- \(n\mu \)
- \(n{X_i}\)
- \(\infty \)
Diketahui |
- Misalkan \({X_1},{X_2}, \ldots ,{X_n}\) suatu variabel acak yang bebas, sehingga setiap \({X_i}\) memiliki “expected value \(\mu \)” dan variansi \({\sigma ^2}\).
- \({S_n} = {X_1} + {X_2} + \cdots + {X_n}\)
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Rumus yang digunakan |
\(E\left[ {{a_1}{X_1} + {a_2}{X_2} + \cdots + {a_n}{X_n}} \right] = {a_1}E\left[ {{X_1}} \right] + {a_2}E\left[ {{X_2}} \right] + \cdots + {a_n}E\left[ {{X_n}} \right]\) |
Proses pengerjaan |
\(E\left[ {{S_n}} \right] = E\left[ {{X_1} + {X_2} + \cdots + {X_n}} \right]\)
\(E\left[ {{S_n}} \right] = E\left[ {{X_1}} \right] + E\left[ {{X_2}} \right] + \cdots + E\left[ {{X_n}} \right]\)
\(E\left[ {{S_n}} \right] = {\mu _1} + {\mu _2} + \cdots + {\mu _n}\)
\(E\left[ {{S_n}} \right] = n\mu \) |
Jawaban |
c. \(n\mu \) |