Suppose that \\(T^{\\ast}(x)\\) and \\(T^{\\ast}(y)\\) are unobserved future lifetimes for \\(x\\) and \\(y\\) that are independent of one another. Let \\(Z\\) be a lifetime random variable that is common to both \\(x\\) and \\(y\\) for e.g., disasters such as earthquakes and hurricanes. Take \\(T^{\\ast}(x)\\), \\(T^{\\ast}(y)\\), \\(Z\\) to be mutually independent.<\/p>\n
Let \\(T(x)= \\min(T^{\\ast}(x),Z)\\) and \\(T(x)= \\min(T^{\\ast}(x),Z)\\) be the observed future lifetimes for \\(x\\) and \\(y\\). Note that they are not independent because they share the common shock random variable \\(Z\\).<\/p>\n
For convenience, assume that the distribution of \\(Z\\) is exponential with constant force \\(\\lambda\\).<\/p>\n
Now, the survival function for \\(x\\) is
\n\\begin{eqnarray*}
\n~_t p_x &=& Pr(T(x) > t) = Pr(\\min(T^{\\ast}(x),Z)>t) \\\\
\n&=& Pr(T^{\\ast}(x)>t) Pr(Z>t) =~ _t p_x^{\\ast} e^{-\\lambda t}.
\n\\end{eqnarray*}
\nSimilarly, \\(~_t p_y = ~_t p_y^{\\ast} e^{-\\lambda t}\\). The joint survival probability is
\n\\begin{eqnarray*}
\n~_t p_{xy} &=& Pr(\\min(T(x),T(y)) > t) = Pr(\\min(T^{\\ast}(x),T^{\\ast}(y),Z)>t) \\\\
\n&=&Pr(T^{\\ast}(x)>t) Pr(T^{\\ast}(y)>t)Pr(Z>t) = ~_t p_x^{\\ast} ~_t p_y^{\\ast} e^{-\\lambda t} \\\\
\n&=& ~_t p_x ~_t p_y e^{\\lambda t} .
\n\\end{eqnarray*}
\nFrom this, we can also calculate the joint force of mortality
\n\\begin{eqnarray*}
\n\\mu_{xy}(t) &=& \\frac{-\\partial }{\\partial t} \\ln ~_t p_{xy} \\\\
\n&=&\\frac{-\\partial}{\\partial t} \\left(\\ln~_t p_x+\\ln ~_t p_y +\\ln e^{\\lambda t} \\right) \\\\
\n&=&\\mu_x(t) + \\mu_y(t) – \\lambda .
\n\\end{eqnarray*} <\/p>\n