Exercises

For a double-decrement model, you are given

  • (mu_{x+t}^{01} = frac{t}{100})
  • (mu_{x+t}^{02} = frac{1}{100})

a) Find the probability of eventually exiting due to cause 2.
Solution.
The probability of surviving is
begin{eqnarray*}
~ _t p_x^{00} &=& exp left{- int_0^t mu_{x+s}^{0bullet} ~ds right} \
&=& exp left{- int_0^t
frac{1+s}{100} ~ds right} =
exp left{-frac{t^2+2t}{200} right} .
end{eqnarray*}
With this, the probability of eventually exiting due to cause 2 is
begin{eqnarray*}
_{infty} p_x^{02} &=&
int_0^{infty} ~ _s p_x^{00} mu_{x+s}^{02} ~ ds = frac{1}{100} int_0^{infty} exp left{-frac{s^2+2s}{200} right} ~ ds \
&=& frac{1}{100} exp left{frac{1}{200} right} int_0^{infty} exp left{-frac{s^2+2s+1}{200} right} ~ ds \
&=& frac{1}{100} exp left{frac{1}{200} right} int_0^{infty} exp left{-frac{(s+1)^2}{200} right} ~ ds \
&=& frac{10}{100} exp left{frac{1}{200} right} int_{0.1}^{infty} exp left{-u^2/2 right} ~ du \
&=& frac{10 sqrt{2 pi}}{100} exp left{frac{1}{200}right} (1 – Phi(0.1)) = 0.1159 ,
end{eqnarray*}
where we use the change of integral (u = (s+1)/sqrt{100}=(s+1)/10) and the cumulative
normal distribution function (Phi(x) = frac{1}{sqrt{2 pi}} int_{-infty}^x e^{-u^2/2} du).
b) Given that an individual has failed due to the second cause, what is the expected time of failure?
Solution.
For part (b), we have
begin{eqnarray*}
frac{1}{ _{infty} p_x^{02}} & & int_0^{infty} t times ~ _t p_x^{00} mu_{x+t}^{02} dt \
&=& frac{1}{0.1159} int_0^{infty} t times exp left{-frac{t^2+2t}{200} right} frac{1}{100} dt \
&=& frac{1}{11.59}exp left{frac{1}{200} right} int_0^{infty} t times exp left{-frac{(t+1)^2}{200} right} dt \
&=& frac{10}{11.59} exp left{frac{1}{200} right} int_{0.1}^{infty} (10u-1) exp left{-u^2/2 right} dt \
&=& frac{10}{11.59}exp left{frac{1}{200}right} 10left. exp left{-u^2/2 right}right|_{0.1}^{infty} \
&~ & ~~~~~~~~ – frac{10sqrt{2 pi}}{11.59}exp left{frac{1}{200}right} (1 -Phi(0.1))=7.63, \
end{eqnarray*}
where we use the change of integral (u =(t+1)/10).

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