Mortality and Survival Models

A life table fragment is given and you must compute t-year survival probability, deferred mortality probability, or expected curtate life.

Read the l_x values directly: t_p_x = l_{x+t} / l_x and t_q_x = 1 - t_p_x. For deferred probability t|u_q_x = t_p_x · u_q_{x+t}. For curtate expectation e_x = Σ_{k=1}^{∞} k_p_x. Use the table's native age progression and don't interpolate unless you must.

t_p_x = l_{x+t} / l_x; e_x = Σ_{k≥1} k_p_x.

A survival function or force of mortality μ(x) is given and you must compute survival probabilities or moments.

Translate between forms: S(x) = exp(-∫₀^x μ(s) ds); μ(x) = -S'(x) / S(x). For t_p_x = S(x + t) / S(x). For complete expectation, ė_x = ∫₀^∞ t_p_x dt. Sketch μ(x) — common parametric forms (Gompertz, Makeham, Weibull) all have characteristic shapes.

μ(x) = f(x) / S(x); S(x) = exp(-∫₀^x μ(s) ds).

A select and ultimate table is given and you must compute probabilities for a select life.

Identify the select period (the columns to the left of "ultimate") and read q_{[x]+s} for s less than the select period; for s beyond, use the ultimate column q_{x+s}. Probabilities accumulate by multiplying p's across the trajectory until you join the ultimate. Recognize that select mortality is lower than ultimate at the same age.

t_p_{[x]} = Π_{k=0}^{t-1} p_{[x]+k} until select period ends.

A model has multiple causes of decrement (e.g., death and lapse), each with its own force of decrement, and you must compute decrement probabilities or expected service.

Write μ^{(j)}(x) for the force of decrement j, and μ^{(τ)}(x) = Σ_j μ^{(j)}(x) for the total force. Then t_p_x^{(τ)} = exp(-∫₀^t μ^{(τ)} ds) and t_q_x^{(j)} = ∫₀^t s_p_x^{(τ)} μ^{(j)}(x + s) ds. Distinguish absolute rates of decrement (single-decrement) from dependent rates (multi-decrement).

t_p_x^{(τ)} = exp(-∫ μ^{(τ)} ds); t_q_x^{(j)} = ∫ s_p_x^{(τ)} μ^{(j)} ds.