Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Misalkan N berdistribusi “negative binomial” dengan E[N] = 20dan Var[N] = 24. Hitunglah nilai dari parameter r
- 5/6
- 20
- 25
- 75
- 100
PEMBAHASAN
| Rumus | \(N\, \sim Negative\,Binomial\,(r\,;p)\) \(E[N] = r\,\frac{q}{p}\, = 20\) \(Var[N] = r\,\frac{q}{{{p^2}}}\, = 24\) |
| Step 1 | \(Var[N] = \left( {r\,\frac{q}{p}} \right)\,\frac{1}{p}\) \(24 = \left( {E[N]} \right)\frac{1}{p}\) \(24 = \left( {20} \right)\frac{1}{p}\) \(p = \frac{5}{6}\) |
| \(q = 1 – p\) \(q = 1 – \frac{5}{6}\) \(q = \frac{1}{6}\) | |
| Step 2 | \(E[N] = r\,\frac{{\left( {\frac{1}{6}} \right)}}{{\left( {\frac{5}{6}} \right)}}\,\) \(20 = r\,\frac{1}{5}\,\) \(r = 100\) |
| Jawaban | e. 100 |