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