The distribution function is
begin{eqnarray*}
F_T(t) &=& Pr(T(overline{xy}) leq t)= Pr(max(T(x), T(y)) leq t) \
&=& Pr(T(x) leq t, T(y) leq t)\
&=&_{IND} Pr(T(x) leq t)times Pr(T(y) leq t) \
&=&(1 – ~_t p_x) times (1- ~_t p_y) .
end{eqnarray*} We write the survivor function as
begin{eqnarray*}
~_t p_{overline{xy}} = 1 – F_T(t) =~_t p_x + ~_t p_y – ~_t p_x ~ ~_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 + ~_t p_y – ~_t p_x ~ ~_t p_y) \
&=&
~_t p_x mu_{x+t} + ~_t p_y mu_{y+t} –
~_t p_x ~_t p_y (mu_{x+t}+mu_{y+t}) \
&=& ~_t p_x mu_{x+t} ~_t q_y +
~_t p_y mu_{y+t} ~_t q_x .
end{eqnarray*} Thus, the force of mortality (mu_{overline{xy}}(t) = frac{f_T(t)}{1-F_T(t)} ) does not have a straightforward expression.