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.
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).
For convenience, assume that the distribution of (Z) is exponential with constant force (lambda).
Now, the survival function for (x) is
begin{eqnarray*}
~_t p_x &=& Pr(T(x) > t) = Pr(min(T^{ast}(x),Z)>t) \
&=& Pr(T^{ast}(x)>t) Pr(Z>t) =~ _t p_x^{ast} e^{-lambda t}.
end{eqnarray*}
Similarly, (~_t p_y = ~_t p_y^{ast} e^{-lambda t}). The joint survival probability is
begin{eqnarray*}
~_t p_{xy} &=& Pr(min(T(x),T(y)) > t) = Pr(min(T^{ast}(x),T^{ast}(y),Z)>t) \
&=&Pr(T^{ast}(x)>t) Pr(T^{ast}(y)>t)Pr(Z>t) = ~_t p_x^{ast} ~_t p_y^{ast} e^{-lambda t} \
&=& ~_t p_x ~_t p_y e^{lambda t} .
end{eqnarray*}
From this, we can also calculate the joint force of mortality
begin{eqnarray*}
mu_{xy}(t) &=& frac{-partial }{partial t} ln ~_t p_{xy} \
&=&frac{-partial}{partial t} left(ln~_t p_x+ln ~_t p_y +ln e^{lambda t} right) \
&=&mu_x(t) + mu_y(t) – lambda .
end{eqnarray*}