Let
- (p_x^{01}) be the probability of attaining Fellowship
- (p_x^{02}) be the probability of exiting the examination system due to other causes.
- Decrements occur at the end of the year
(x) | (p_x^{01}) | (p_x^{02}) |
21 | 0.008 | 0.015 |
22 | 0.015 | 0.20 |
23 | 0.025 | (_t q_x^{(tau)}) |
After attaining fellowship, the only decrement is mortality with constant force (mu = 0.04.)
Calculate the probability that a student age 21 will be living and a Fellow 3 years later.
Solution.
(x) | (p_x^{01}) | (p_x^{02}) | (p_x^{0bullet}) | (p_x^{00}) | (_n p_x^{00}) | ( n) |
21 | 0.008 | 0.015 | 0.158 | 0.842 | 0.842 | 1 |
22 | 0.015 | 0.20 | 0.215 | 0.785 | 0.661 | 2 |
23 | 0.025 | 0.25 | 0.275 | 0.725 | 0.479 |
begin{eqnarray*}
&&text{Desired probability is}\
&=& p_{21}^{01} p_{22} p_{23} + p_{21}^{00} p_{22}^{01} p_{23} + ~_2p_{21}^{00} p_{23}^{01} \
&=&(0.008) e^{-0.04} e^{-0.04}+ (0.842)(.015) e^{-0.04} + (0.661)(0.025) = 0.036
end{eqnarray*}