Suppose that \\(T(x)\\) is exponentially distributed with force of mortality \\(\\mu_x(t)=\\mu_1\\), \\(T(y)\\) is exponentially distributed with force of mortality \\(\\mu_y(t)=\\mu_2\\), and that \\(T(x)\\) and \\(T(y)\\) are independent. Then, for \\(x\\), we have \\(~_t p_x = \\exp(-\\mu_1 t)\\) and
\n\\begin{eqnarray*}
\n\\dot{e}_x &=& E ~ T(x) = \\int_0^{\\infty} ~_t p_x ~ dt =
\n\\frac{1}{\\mu_1} .
\n\\end{eqnarray*} Similarly, for the joint-life status, we have
\n\\(~_t p_{xy} = ~_t p_{xy} \\times ~_t p_{xy} = \\exp(-(\\mu_1+\\mu_2) t)\\) which is an exponential distribution with force of mortality \\(\\mu_1+\\mu_2\\). This yields
\n\\begin{eqnarray*}
\n\\dot{e}_{xy}=\\frac{1}{\\mu_1+\\mu_2} .
\n\\end{eqnarray*} Now, for the last-survivor status, we have
\n\\begin{eqnarray*}
\n1 – ~_t p_{\\overline{xy}} &=& F_{T(\\overline{xy})}(t) = \\Pr\\left(
\n\\max(T(x),T(y)) \\leq t \\right) \\\\
\n&=& \\Pr\\left( T(x) \\leq t \\right) \\times \\Pr\\left( T(y) \\leq t \\right) \\\\
\n&=& \\left( 1-\\exp(-\\mu_1 t) \\right) \\times \\left( 1-\\exp(-\\mu_2 t)\\right)
\n\\end{eqnarray*} and
\n\\begin{eqnarray*}
\n\\dot{e}_{\\overline{xy}} &=& \\dot{e}_{x}+\\dot{e}_{y}-\\dot{e}_{xy}\\\\
\n&=& \\frac{1}{\\mu_1} +\\frac{1}{\\mu_2} -\\frac{1}{\\mu_1+\\mu_2} .
\n\\end{eqnarray*}<\/p>\n
Follow-up<\/strong>
\nSuppose in addition that \\(\\mu_1 = 0.02\\) and \\(\\mu_2 = 0.015\\). Then,
\n\\begin{eqnarray*}
\n\\dot{e}_x = \\frac{1}{\\mu_1} = 50,~~~~\\dot{e}_y = \\frac{1}{\\mu_2} =
\n66.67, ~~~\\dot{e}_{xy} = \\frac{1}{\\mu_1+\\mu_2} = 28.57 .
\n\\end{eqnarray*} Further,
\n\\begin{eqnarray*}
\n\\dot{e}_{\\overline{xy}} &=& \\dot{e}_{x}+\\dot{e}_{y}-\\dot{e}_{xy}\\
\n&=& 50 + 66.67 -28.57 = 88.1 .
\n\\end{eqnarray*} Moreover, we have
\n\\begin{eqnarray*}
\nCov(T(\\overline{xy}),T(xy)) &=& \\mathrm{E~} \\left\\{T(\\overline{xy})\\times T(xy)\\right\\} – \\mathrm{E~} T(\\overline{xy}) \\times
\n\\mathrm{E~}T(xy)\\\\
\n&=& \\mathrm{E~} \\left\\{\\max(T(x),T(y)) \\times \\min (T(x),T(y))\\right\\} – \\dot{e}_{\\overline{xy}} \\times \\dot{e}_{xy}\\\\
\n&=& \\mathrm{E~}\\left\\{ T(x) \\times T(y)\\right\\}
\n– \\dot{e}_{\\overline{xy}} \\times \\dot{e}_{xy}\\\\
\n&=& \\dot{e}_x \\dot{e}_y – \\dot{e}_{\\overline{xy}} \\times
\n\\dot{e}_{xy} \\\\
\n&=& 50 \\times 66.67 – 88.1 \\times 28.57 = 816.48 .
\n\\end{eqnarray*}
\n