The density function f_x(t) for a life aged x in continuous time is related to tpx and the force of mortality μ_{x+t}. Which expression correctly represents f_x(t)?

• A. f_x(t) = tpx * μ_{x+t}
• B. f_x(t) = μ_{x+t} / tpx
• C. f_x(t) = tpx + μ_{x+t}
• D. f_x(t) = S(x+t)

Answer: (A)

Think of the time to death T for a life aged x in terms of hazard and survival. The density f_x(t) is the probability per unit time that death occurs in (t, t+dt), given the person has survived to age x+t. To have a death in that interval, the person must have reached time t alive (probability {}{t}p_x, or tpx) and then die at rate μ{x+t} over the small interval dt (probability ≈ μ_{x+t} dt). Multiplying these together gives f_x(t) dt ≈ {}{t}p_x μ{x+t} dt, so the density is f_x(t) = {}{t}p_x μ{x+t}. Since {}{t}p_x is tpx, the expression is f_x(t) = tpx · μ{x+t}.

This matches the idea that the instantaneous mortality hazard at age x+t, scaled by the probability of having survived to that age, yields the density of the time of death. The other forms don’t align with how a density is constructed from survival and hazard: they mix terms incorrectly or use the survival function itself rather than the hazard-weighted survival.