Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Probabilitas dan Statistik |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
8 |
SOAL
Misal X adalah sebuah variable acak dengan “moment generating function” \(M(t) = {\left( {\frac{{2 + {e^t}}}{3}} \right)^9}\). Hitunglah variansi dari X
- 2
- 3
- 8
- 9
- 11
| Step 1 |
\(E[X] = {\left. {\frac{{\partial M(t)}}{{\partial t}}} \right|_{t = o}}\)
\(E[X] = \frac{\partial }{{\partial t}}{\left. {{{\left( {\frac{{2 + {e^t}}}{3}} \right)}^9}} \right|_{t = 0}}\)
\(E[X] = (9)\frac{{{e^t}}}{3}{\left. {{{\left( {\frac{{2 + {e^t}}}{3}} \right)}^8}} \right|_{t = 0}}\)
\(E[X] = \frac{9}{3}\,\)
\(E[X] = 3\) |
| Step 2 |
\(E[{X^2}] = {\left. {\frac{{\partial M'(t)}}{{\partial t}}} \right|_{t = o}}\)
\(E[{X^2}] = \frac{\partial }{{\partial t}}(9)\frac{{{e^t}}}{3}{\left. {{{\left( {\frac{{2 + {e^t}}}{3}} \right)}^8}} \right|_{t = 0}}\)
\(E[{X^2}] = (9)\frac{{{e^t}}}{3}{\left. {{{\left( {\frac{{2 + {e^t}}}{3}} \right)}^8} + (9)\frac{{{e^t}}}{3}(8)\left( {\frac{{{e^t}}}{3}} \right){{\left( {\frac{{2 + {e^t}}}{3}} \right)}^7}} \right|_{t = 0}}\)
\(E[{X^2}] = \frac{9}{3}\, + \left( {\frac{9}{3}\frac{8}{3}} \right)\)
\(E[{X^2}] = 11\) |
| Step 3 |
\(Var[X] = E[{X^2}] – E{[X]^2}\)
\(Var[X] = 11 – {3^2}\)
\(Var[X] = 2\) |
| Jawaban |
a. 2 |
