The distribution function of (T=T(xy)) is
begin{eqnarray*}
F_T(t) &=& Pr left(T(xy) leq t right) \ &=&Pr(min(T(x), T(y)) leq t)\
&=& 1- Pr(min(T(x), T(y)) gt t) \
&=& 1- Pr(T(x)gt t) times Pr(T(y) gt t) \
&=& 1~-~_t p_x times ~_t p_y .
end{eqnarray*} We write the survivor function as
begin{eqnarray*}
~_t p_{xy} = 1 – F_T(t) = ~_t p_x times ~_t p_y .
end{eqnarray*} From this, the density function is
begin{eqnarray*}
f_T(t) &=& F^{prime}_T(t) = – frac{partial}{partial t} (~_t p_x times ~_t p_y) \
&=& – left( ~_t p_x frac{partial}{partial t} ~_t p_y +
~_t p_y frac{partial}{partial t} ~_t p_x right) \
&=& ~_t p_x ( ~_t p_y mu_{y+t} )+
~_t p_y (~_t p_x mu_{x+t} ) \
&=& ~_t p_x ~_t p_y (mu_{x+t}+mu_{y+t}) .
end{eqnarray*} Thus, the force of mortality is
begin{eqnarray*}
mu_{xy}(t) = frac{f_T(t)}{1-F_T(t)} = mu_{x+t}+mu_{y+t} .
end{eqnarray*}