Which expression represents f(x) in terms of other quantities?

• A. p_x · μ_x
• B. tpx · μ_x
• C. p_x · μ_{x+t}
• D. tpx · μ_{x+t}

Answer: (D)

The density f_x(t) of the remaining lifetime T for a person aged x is given by the survival up to time t times the hazard at time t. Specifically, S_x(t) = tpx is the probability of surviving t years from age x, and the instantaneous force of mortality at age x+t is μ_{x+t}. The density of death at time t is f_x(t) = μ_{x+t} · S_x(t) = tpx · μ_{x+t}. So the correct expression combines the future-age hazard with the survival to that future time, which is exactly tpx · μ_{x+t}. The other options mix different quantities that do not reflect the density at time t.