Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Untuk T , variabel acak “future lifetime” pada (0), diberikan sebagai berikut:
- \(\omega > 70\)
- \({}_{40}{P_0} = 0,6\)
- \(E[T] = 62\)
- \(E[min(T,t)] = t – 0,005{t^2},\) \(0 < T < 60\)
Hitunglah “complete expectation of life” pada 40
- 30
- 35
- 40
- 45
- 50
| Step 1 | \(E[min(T,t)] = t – 0,005{t^2}\)
\(E[min(T,40)] = 40 – 0,005{(40)^2}\)
\(E[min(T,40)] = 32\)
|
| Step 2 | \(E[T] = \int_0^{40} {{}_x{p_0}dx} + \int_{40}^\omega {{}_x{p_0}dx} \) \(62 = 32 + \int_{40}^\omega {{}_x{p_0}dx} \) \(\int_{40}^\omega {{}_x{p_0}dx} = 30\) |
| Step 3 | \({\mathop e\limits^o _{40}} = \frac{{\int_{40}^\omega {{}_x{p_0}dx} }}{{{}_{40}{p_0}}}\) \({\mathop e\limits^o _{40}} = \frac{{30}}{{0,6}}\) \({\mathop e\limits^o _{40}} = 50\) |
| Jawaban | e. 50 |