For a double-decrement model, you are given
- (mu_{x+t}^{01} = frac{r_1}{c} t)
- (mu_{x+t}^{02} = frac{r_2}{c} t)
Determine the probability of eventually exiting due to cause 1.
Solution.
The probability of eventually exiting due to cause 1 may be expressed as
begin{eqnarray*}
_{infty} p_x^{01}=
int_0^{infty} ~ _s p_x^{00} mu_{x+s}^{01} ~ ds = frac{r_1}{c} int_0^{infty} ~ _s p_x^{00} s ~ ds .
end{eqnarray*}
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{r_1+r_2}{c} s ~ds right} =
exp left{-frac{r_1+r_2}{2c} t^2 right} .
end{eqnarray*}
With this, the probability of eventually exiting due to cause 1 is
begin{eqnarray*}
_{infty} p_x^{01} &=& frac{r_1}{c} int_0^{infty} ~ exp left{-frac{r_1+r_2}{c} frac{s^2}{2} right} s ~ ds \
&=& left. -frac{r_1}{c} frac{1}{frac{r_1+r_2}{c}}exp left{-frac{r_1+r_2}{c}
frac{t^2}{2} right} right|_0^{infty} = frac{r_1}{r_1+r_2} .
end{eqnarray*}