The age at death \(X\) of a new-born is the continuous random variable representing the total length of life. It takes values in \([0, \omega)\) where \(\omega\) is the limiting age (typically taken as 110 or 120).
The CDF of \(X\) is \(F_X(x) = P(X \le x)\) — probability of dying by age \(x\). The PDF \(f_X(x)\) is the rate of death at age \(x\).
The survival function is the probability that a new-born survives beyond age \(x\):
\[ s(x) \;=\; P(X > x) \;=\; 1 - F_X(x). \]For a person aged \(x\) (who is alive), \(T(x) = X - x\) is the remaining future lifetime, a positive continuous r.v. on \([0, \omega - x)\).
| Symbol | Meaning | Formula |
|---|---|---|
| \(_tp_x\) | Prob. (\(x\)) survives \(t\) more years | \(_tp_x = s(x+t)/s(x)\) |
| \(_tq_x\) | Prob. (\(x\)) dies within \(t\) years | \(_tq_x = 1 - {}_tp_x\) |
| \(_{s|t}q_x\) | Prob. (\(x\)) survives \(s\) yrs, then dies within next \(t\) yrs | \(_{s|t}q_x = {}_sp_x - {}_{s+t}p_x\) |
| \(p_x\) | Prob. (\(x\)) survives 1 more year | \(p_x = s(x+1)/s(x)\) |
| \(q_x\) | Prob. (\(x\)) dies within 1 year | \(q_x = 1 - p_x\) |
where \(\mu(x+t)\) is the force of mortality at age \(x+t\) — see Section 5.
The curtate future lifetime \(K(x) = \lfloor T(x) \rfloor\) is the integer part of \(T(x)\) — the number of complete future years the life will live.
\(K(x)\) is a discrete r.v. taking values \(0, 1, 2, \ldots\) with PMF:
\[ P(K(x) = k) \;=\; {}_kp_x \cdot q_{x+k} \;=\; {}_{k|}q_x. \]This is the average number of complete future years of life — a standard life-table column.
Under the uniform-distribution-of-deaths assumption (Section 7): \(\overset{\circ}{e}_x \approx e_x + 1/2\).
The force of mortality at age \(x\) is the instantaneous rate of death given survival to age \(x\) — the continuous analogue of \(q_x\):
\[ \mu(x) \;=\; -\dfrac{s'(x)}{s(x)} \;=\; \dfrac{f_X(x)}{s(x)}. \]The survival function can be recovered from the force of mortality:
If \(\mu(x) = \mu\) constant, then \(s(x) = e^{-\mu x}\) and \(X \sim\) Exponential(\(\mu\)).
\(_tp_x = e^{-\mu t}\) is independent of \(x\) — the memoryless property.
\(E(T(x)) = 1/\mu\). If \(\mu = 0.02\), expected lifetime = 50 years from any age.
De Moivre's law: \(s(x) = 1 - x/\omega\) for \(0 \le x \le \omega\); deaths uniform on \([0, \omega]\).
\(f(x) = 1/\omega\); \(\mu(x) = 1/(\omega - x)\) — force increases with age.
If \(\omega = 100\), \(s(60) = 0.4\); \(\mu(60) = 1/40 = 0.025\).
A life table is a tabular representation of mortality experience, showing for each integer age \(x\) the expected number of survivors out of a starting cohort. It is the central data object in life-contingencies work.
| Symbol | Meaning | Recursion |
|---|---|---|
| \(l_x\) | Expected number alive at exact age \(x\) (starting cohort \(l_0\) = radix, usually 100 000) | — |
| \(d_x\) | Expected deaths between ages \(x\) and \(x+1\) | \(d_x = l_x - l_{x+1}\) |
| \(q_x\) | Probability life aged \(x\) dies within 1 year | \(q_x = d_x/l_x\) |
| \(p_x\) | Probability life aged \(x\) survives 1 year | \(p_x = l_{x+1}/l_x = 1 - q_x\) |
| \(L_x\) | Person-years lived between ages \(x\) and \(x+1\) | \(\approx (l_x + l_{x+1})/2\) |
| \(T_x\) | Total future person-years from age \(x\) onward | \(T_x = \sum_{y \ge x} L_y\) |
| \(e_x^{\circ}\) | Complete expectation of life at \(x\) | \(e_x^{\circ} = T_x/l_x\) |
Imagine a large cohort of \(l_0\) lives, each subject to the same mortality. By the law of large numbers, the actual number of survivors at age \(x\) is approximately \(l_x = l_0 \cdot s(x)\). This is the deterministic survivorship group interpretation that justifies treating \(l_x\) as a function of \(x\) rather than a random count.
If \(s(x) = (100 - x)/100\) (De Moivre with ω = 100), \(l_0 = 100{,}000\):
\(l_{30} = 100000 \cdot (70/100) = 70{,}000\); \(l_{60} = 40{,}000\); \(l_{90} = 10{,}000\).
\(q_{60} = (l_{60} - l_{61})/l_{60} = (40000 - 39000)/40000 = 0.025\).
\(_{10}p_{60} = l_{70}/l_{60} = 30000/40000 = 0.75\).
For De Moivre with ω = 100, \(_kp_{30} = (70 - k)/70\). Then:
\(e_{30} = \sum_{k=1}^{69} (70-k)/70 = (1/70)(69 + 68 + \cdots + 1) = (1/70)(69 \cdot 70/2) = 34.5\) years.
Complete expectation: \(e_{30}^{\circ} \approx e_{30} + 1/2 = 35\) years (matches \((\omega - 30)/2 = 35\) for De Moivre).
Life tables give us values at integer ages. To compute probabilities at fractional ages (e.g. \(_{0.5}p_{30}\)) we need an interpolation assumption. The three standard ones:
Within each year \([x, x+1]\), deaths are spread uniformly over the year.
So \(s(x + s) = s(x) - s \cdot d_x / l_0\) interpolates linearly. Force of mortality \(\mu(x+s) = q_x / (1 - s\, q_x)\).
Within each year, the force of mortality is constant. Let \(\mu_x\) be that constant; then \(p_x = e^{-\mu_x}\), so \(\mu_x = -\ln p_x\).
Survival function is hyperbolic within the year: \(_{1-s}q_{x+s} = (1-s) q_x\).
| Assumption | μ within year | Easiest formula |
|---|---|---|
| UDD | Increasing | Linear in s |
| Constant force | Constant | Exponential in s |
| Balducci | Decreasing | Hyperbolic |
If \(q_{50} = 0.005\), find \(_{0.5}p_{50}\) under UDD.
\(_{0.5}p_{50} = 1 - 0.5 \cdot 0.005 = 0.9975\).
Same \(q_{50} = 0.005\). Under constant force: \(\mu_{50} = -\ln 0.995 = 0.005013\). \(_{0.5}p_{50} = 0.995^{0.5} = 0.99749\).
Values match closely for small \(q\), but diverge for large \(q\) (older ages).
Parametric formulas for \(\mu(x)\) or \(s(x)\) used to smooth life-table data or extrapolate to old ages.
The most influential — describes mortality from young adulthood onwards.
The log of mortality is linear in age. Survival:
\[ s(x) \;=\; \exp\!\left(-\dfrac{B(c^x - 1)}{\ln c}\right). \]Adds a constant term \(A\) capturing age-independent risks (accidents, infections):
Widely used in classical actuarial work to graduate (smooth) the mortality rates of a life table.
Used in reliability engineering and biological aging.
Heligman–Pollard, Coale–Demeny, log-quadratic mortality models — refinements used by demographers and actuaries today.
If \(B = 0.0001\), \(c = 1.1\): \(\mu(50) = 0.0001 \cdot (1.1)^{50} = 0.0001 \cdot 117.4 = 0.01174\).
Mortality at 70: \(\mu(70) = 0.0001 \cdot (1.1)^{70} = 0.0001 \cdot 789.7 = 0.07897\) — ~7 times higher than at 50. The exponential nature explains rapid aging.
With \(A = 0.0005\) added: \(\mu(20) = 0.0005 + 0.0001 \cdot 1.1^{20} = 0.0005 + 0.000673 = 0.001173\). Even at age 20 the Gompertz term (0.000673) is slightly larger than \(A\), but \(A\) is still 43% of the total. Its share keeps shrinking with age as \(Bc^x\) grows, so \(A\) matters mainly at young ages and the Gompertz term drives mortality at older ages.