The distribution function is
\n\\begin{eqnarray*}
\nF_T(t) &=& \\Pr(T(\\overline{xy}) \\leq t)= \\Pr(\\max(T(x), T(y)) \\leq t) \\\\
\n&=& \\Pr(T(x) \\leq t, T(y) \\leq t)\\\\
\n&=&_{IND} \\Pr(T(x) \\leq t)\\times Pr(T(y) \\leq t) \\\\
\n&=&(1 – ~_t p_x) \\times (1- ~_t p_y) .
\n\\end{eqnarray*} We write the survivor function as
\n\\begin{eqnarray*}
\n~_t p_{\\overline{xy}} = 1 – F_T(t) =~_t p_x + ~_t p_y – ~_t p_x ~ ~_t p_y .
\n\\end{eqnarray*} From this, the density function is
\n\\begin{eqnarray*}
\nf_T(t) &=& F^{\\prime}_T(t) = – \\frac{\\partial}{\\partial t} (~_t p_x + ~_t p_y – ~_t p_x ~ ~_t p_y) \\\\
\n&=&
\n~_t p_x \\mu_{x+t} + ~_t p_y \\mu_{y+t} –
\n~_t p_x ~_t p_y (\\mu_{x+t}+\\mu_{y+t}) \\\\
\n&=& ~_t p_x \\mu_{x+t} ~_t q_y +
\n~_t p_y \\mu_{y+t} ~_t q_x .
\n\\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.<\/p>\n