Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2014 |
| Nomor Soal | : | 22 |
SOAL
Sekumpulan dari \(n\) orang diamati sampai semuanya meninggal, dengan kematian dikelompokkan dalam interval yang tetap. Jika \(Var\left[ {\hat S\left( t \right)} \right] = 0,0009\) , \(Var\left[ {\hat S\left( r \right)} \right] = 0,0016\) , dan covarians-nya adalah \(Cov\left[ {\hat S\left( t \right),\hat S\left( r \right)} \right] = 0,0008\) . Tentukan \(E\left[ {\hat S\left( t \right)} \right]\)
- 0,6
- 0,7
- 0,8
- 0,9
- Tidak ada jawaban yang benar
| Diketahui | - Sekumpulan dari \(n\) orang diamati sampai semuanya meninggal, dengan kematian dikelompokkan dalam interval yang tetap.
- \(Var\left[ {\hat S\left( t \right)} \right] = 0,0009\)
- \(Var\left[ {\hat S\left( r \right)} \right] = 0,0016\)
- covarians-nya adalah \(Cov\left[ {\hat S\left( t \right),\hat S\left( r \right)} \right] = 0,0008\)
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| Rumus yang digunakan | - \(E\left[ {\hat S\left( t \right)} \right] = S\left( t \right)\)
- \(Var\left[ {\hat S\left( t \right)} \right] = \frac{{S\left( t \right)F\left( t \right)}}{n} = \frac{{S\left( t \right)\left( {1 – S\left( t \right)} \right)}}{n}\)
- \(Cov\left[ {\hat S\left( t \right),\hat S\left( r \right)} \right] = \frac{{F\left( t \right)S\left( r \right)}}{n}\)
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| Proses pengerjaan | - \(Cov\left[ {\hat S\left( t \right),\hat S\left( r \right)} \right] = \frac{{F\left( t \right)S\left( r \right)}}{n} \Leftrightarrow 0.0008n = F\left( t \right)S\left( r \right)\)
- \(Var\left[ {\hat S\left( t \right)} \right] = \frac{{S\left( t \right)F\left( t \right)}}{n} \Leftrightarrow 0.0009n = S\left( t \right)F\left( t \right)\)
- \(Var\left[ {\hat S\left( r \right)} \right] = \frac{{S\left( r \right)F\left( r \right)}}{n} \Leftrightarrow 0.0016n = S\left( r \right)F\left( r \right)\)
- \(\frac{{F\left( t \right)S\left( r \right)}}{{0.0008}} = \frac{{S\left( r \right)F\left( r \right)}}{{0.0016}} \Leftrightarrow \frac{{F\left( t \right)}}{{F\left( r \right)}} = \frac{8}{{16}}\)
- \(\frac{{F\left( t \right)S\left( r \right)}}{{0.0008}} = \frac{{S\left( t \right)F\left( t \right)}}{{0.0009}} \Leftrightarrow \frac{{S\left( t \right)}}{{S\left( r \right)}} = \frac{9}{8}\)
- \(Cov\left[ {\hat S\left( t \right),\hat S\left( r \right)} \right] = \frac{{F\left( t \right)S\left( r \right)}}{n}\)
\(0.0008n = \left[ {\frac{{0.0009n}}{{S\left( t \right)}}} \right] \cdot \left[ {\frac{{0.0016n}}{{F\left( r \right)}}} \right]\)
\(S\left( t \right)F\left( r \right) = \frac{9}{8}\left( {0.0016n} \right)\)
\(S\left( t \right)F\left( r \right) = 0.0018n\)
\(\left[ {1 – F\left( t \right)} \right]\left[ {1 – S\left( r \right)} \right] = 0.0018n\)
\(1 – F\left( t \right) – S\left( r \right) + F\left( t \right)S\left( r \right) = 0.0018n\)
\(S\left( t \right) – \frac{8}{9}S\left( t \right) + 0.0008n = 0.0018n\)
\(S\left( t \right) = 9\left( {0.0018n – 0.0008n} \right)\)
\(S\left( t \right) = 0.009n\)
- \(0.0009n = S\left( t \right)F\left( t \right)\)
\(0.0009n = 0.009n\left( {1 – 0.009n} \right)\)
\(n = \frac{{0.009n – \left( {0.009n} \right)\left( {0.009n} \right)}}{{0.0009}}\)
\(n = 10n – 0.09{n^2}\)
\(0.09{n^2} = 9n\)
\(n = \frac{9}{{0.09}} = 100\)
- \(E\left[ {\hat S\left( t \right)} \right] = S\left( t \right) = 0.009n = 0.009\left( {100} \right) = 0.9\)
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| Jawaban | D. 0,9 |
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