Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
3 |
SOAL
Diketahui \({l_x} = 2.500{(64 – 0,8x)^{\frac{1}{3}}},\,0 \le x \le 80\). Tentukanlah angka yang paling dekat untuk \(Var[X] – {(E[X])^2}\).
- 60
- 514
- 3.600
- 4.114
- 7.714
| Step 1 |
\({l_0}\, = 2.500{(64 – 0,8(0))^{\frac{1}{3}}}\)
\({l_0}\, = 10.000\) |
| Step 2 |
\(E[X]\, = \int\limits_0^{80} {\frac{{{l_{0 + t}}}}{{{l_0}}}} dt\)
\(E[X] = \,\frac{{\int\limits_0^{80} {2.500{{(64 – 0,8x)}^{\frac{1}{3}}}} }}{{10.000}}\,\)
\(E[X] = \,\frac{{2.500\frac{3}{4}\left( {\frac{{10}}{8}} \right)\left[ {{{(64 – 0,8(0))}^{\frac{4}{3}}} – {{(64 – 0,8(80))}^{\frac{4}{3}}}} \right]}}{{10.000}}\)
\(E[X] = \frac{{600.000}}{{10.000}}\)
\(E[X] = 60\) |
| Step 3 |
\(E[{X^2}]\,\, = 2\int\limits_0^{80} {x\frac{{{l_{0 + t}}}}{{{l_0}}}} dt\,\)
\(E[{X^2}] = \,\frac{{2\int\limits_0^{80} {x\,\,\left[ {2.500{{(64 – 0,8x)}^{\frac{1}{3}}}} \right]} dt}}{{10.000}}\,\,\)
\(E[{X^2}] = \,4.114,285711\) |
| Step 4 |
\(Var[X] = E[{X^2}]\,\, – \,E{[X]^2}\)
\(Var[X] = 4.114 – {60^2}\)
\(Var[X] = 514\) |
| Maka |
\(Var[X] – E{[X]^2} = 514 – {60^2}\)
\(Var[X] – E{[X]^2} = – 3.086\) |
| Jawaban |
Anulir |
