Survival Models and Life Table Functions
This domain tests how future lifetime is modelled as a random variable and how life-table columns convert survival models into probabilities, expected times and annuity factors.
Core relationships
- For a life aged x, T_x is the future lifetime and K_x = floor(T_x) is the curtate future lifetime. The survival function is S_x(t) = P(T_x > t).
- At integer ages, S_x(t) = l_{x+t}/l_x, so p_x = l_{x+1}/l_x and q_x = d_x/l_x = 1 - p_x.
- The force of mortality is mu_{x+t} = -d/dt log S_x(t), giving S_x(t) = exp(-integral_0^t mu_{x+s} ds).
- Life-table identities: d_x = l_x - l_{x+1}; m_x = d_x/L_x; e_x = sum_{k>=1} k p_x; e_x-degrees = integral_0^infinity t p_x dt.
- Fractional-age assumptions matter. Under uniform distribution of deaths (UDD), t q_x = t q_x for 0 < t < 1, so l_{x+t} = l_x - t d_x. Under constant force, p_x over a fraction t is (p_x)^t. Under Balducci, 1/l_{x+t} is linear.
Worked example
Suppose l_60 = 90,000 and l_61 = 89,400. Then q_60 = 600/90,000 = 0.006667 and p_60 = 0.993333. Under UDD, the probability that a life aged 60 dies within the first three months is 0.25 q_60 = 0.001667. The corresponding force is not q_60: under constant force, mu = -log(p_60) = 0.006689, so the three-month death probability would be 1 - exp(-0.25 mu) = 0.001669. The two assumptions give close but different answers.
Traps that cost marks
Do not confuse q_x with m_x or mu_x. Do not use e_x when the question asks for the complete expectation e_x-degrees. Never apply UDD over a whole year when the question specifies constant force, and do not treat l_x as a probability scale: only ratios such as l_{x+t}/l_x are meaningful.