For a double-decrement model, you are given<\/p>\n
a) Find the probability of eventually exiting due to cause 2.
\nSolution.<\/em>
\nThe probability of surviving is
\n\\begin{eqnarray*}
\n~ _t p_x^{00} &=& \\exp \\left\\{- \\int_0^t \\mu_{x+s}^{0\\bullet} ~ds \\right\\} \\\\
\n&=& \\exp \\left\\{- \\int_0^t
\n\\frac{1+s}{100} ~ds \\right\\} =
\n\\exp \\left\\{-\\frac{t^2+2t}{200} \\right\\} .
\n\\end{eqnarray*}
\nWith this, the probability of eventually exiting due to cause 2 is
\n\\begin{eqnarray*}
\n_{\\infty} p_x^{02} &=&
\n\\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 \\\\
\n&=& \\frac{1}{100} \\exp \\left\\{\\frac{1}{200} \\right\\} \\int_0^{\\infty} \\exp \\left\\{-\\frac{s^2+2s+1}{200} \\right\\} ~ ds \\\\
\n&=& \\frac{1}{100} \\exp \\left\\{\\frac{1}{200} \\right\\} \\int_0^{\\infty} \\exp \\left\\{-\\frac{(s+1)^2}{200} \\right\\} ~ ds \\\\
\n&=& \\frac{10}{100} \\exp \\left\\{\\frac{1}{200} \\right\\} \\int_{0.1}^{\\infty} \\exp \\left\\{-u^2\/2 \\right\\} ~ du \\\\
\n&=& \\frac{10 \\sqrt{2 \\pi}}{100} \\exp \\left\\{\\frac{1}{200}\\right\\} (1 – \\Phi(0.1)) = 0.1159 ,
\n\\end{eqnarray*}
\nwhere we use the change of integral \\(u = (s+1)\/\\sqrt{100}=(s+1)\/10\\) and the cumulative
\nnormal distribution function \\(\\Phi(x) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^x e^{-u^2\/2} du\\).
\nb) Given that an individual has failed due to the second cause, what is the expected time of failure?
\nSolution.<\/em>
\nFor part (b), we have
\n\\begin{eqnarray*}
\n\\frac{1}{ _{\\infty} p_x^{02}} & & \\int_0^{\\infty} t \\times ~ _t p_x^{00} \\mu_{x+t}^{02} dt \\\\
\n&=& \\frac{1}{0.1159} \\int_0^{\\infty} t \\times \\exp \\left\\{-\\frac{t^2+2t}{200} \\right\\} \\frac{1}{100} dt \\\\
\n&=& \\frac{1}{11.59}\\exp \\left\\{\\frac{1}{200} \\right\\} \\int_0^{\\infty} t \\times \\exp \\left\\{-\\frac{(t+1)^2}{200} \\right\\} dt \\\\
\n&=& \\frac{10}{11.59} \\exp \\left\\{\\frac{1}{200} \\right\\} \\int_{0.1}^{\\infty} (10u-1) \\exp \\left\\{-u^2\/2 \\right\\} dt \\\\
\n&=& \\frac{10}{11.59}\\exp \\left\\{\\frac{1}{200}\\right\\} 10\\left. \\exp \\left\\{-u^2\/2 \\right\\}\\right|_{0.1}^{\\infty} \\\\
\n&~ & ~~~~~~~~ – \\frac{10\\sqrt{2 \\pi}}{11.59}\\exp \\left\\{\\frac{1}{200}\\right\\} (1 -\\Phi(0.1))=7.63, \\\\
\n\\end{eqnarray*}
\nwhere we use the change of integral \\(u =(t+1)\/10\\). <\/p>\n