Example: Students Writing Actuarial Exams

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*}

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