Life Tables: Mortality Rates, Survival Functions, and Notation

Core Notation

The life table is the foundation of life contingencies. Key functions include l_x (number of lives at age x), d_x = l_x - l_{x+1} (deaths between ages x and x+1), q_x = d_x/l_x (one-year mortality probability), and p_x = 1 - q_x (one-year survival probability). Multi-year probabilities use the notation: t_p_x = l_{x+t}/l_x is the probability that (x) survives t years, and t_q_x = 1 - t_p_x is the probability of death within t years. The deferred mortality probability t|u_q_x = t_p_x minus (t+u)_p_x gives the probability of death between times t and t+u.

Survival and Force of Mortality

The survival function S(x) = P(T_0 > x) gives the probability of surviving to age x from birth. The force of mortality mu_x = -S'(x)/S(x) = -d/dx ln(S(x)) is the instantaneous rate of mortality at age x. It connects to discrete mortality via p_x = exp(-integral from 0 to 1 of mu_{x+t} dt). Common parametric mortality laws include Gompertz (mu_x = Bc^x) and Makeham (mu_x = A + Bc^x). Exam LTAM requires fluency with these relationships for computing insurance and annuity values.