Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Probabilita dan Statistika |
| Periode Ujian |
: |
Juni 2014 |
| Nomor Soal |
: |
12 |
SOAL
Diketahui variabel acak X berdistribusi Poisson dengan mean \(\lambda \) . Jika Pr( X = 1 | X \(\le \) 1) = 80%, maka nilai dari \(\lambda \) sama dengan …
- 4
- – ln (2)
- 0,80
- 0,25
- – ln (0,8)
| Diketahui |
Variabel acak X berdistribusi Poisson dengan mean \(\lambda \)
Jika Pr( X = 1 | X \(\le \) 1) = 80% |
| Rumus yang digunakan |
\(\Pr \left( {X = 1\left| {X \le 1)} \right.} \right) = \frac{{\Pr (X = 1)}}{{\Pr (X \le 1)}}\) |
| Proses pengerjaan |
\({f_X}(x) = {e^{ – \lambda }}\frac{{{\lambda ^x}}}{{x!}},x = 0,1,2,…\)
\(Pr(X = 1|X \le 1){\rm{ }} = {\rm{ }}80\% = 0,8\)
\(\Pr \left( {X = 1\left| {X \le 1)} \right.} \right) = \frac{{\Pr (X = 1)}}{{\Pr (X \le 1)}} = 0,8\)
\(\Leftrightarrow \frac{{{e^{ – \lambda }}\frac{{{\lambda ^1}}}{{1!}}}}{{{e^{ – \lambda }}\frac{{{\lambda ^0}}}{{0!}} + {e^{ – \lambda }}\frac{{{\lambda ^1}}}{{1!}}}} = 0,8\)
\(\Leftrightarrow \frac{\lambda }{{1 + \lambda }} = 0,8\)
\(\Leftrightarrow \lambda = 4\) |
| Jawaban |
a. 4 |
