Parametric Survival Models, Life Tables, and Their Relationships

Problem Preview: A population follows Gompertz's law with $B = 0.0003$ and $c = 1.07$. You need ${}_{10}p_{50}$. Half the candidates write the integral of $Bc^{50+s}$ as $Bc^{50} \cdot 10$, treating the exponential as constant. The correct integral is $Bc^{50}(c^{10}-1)/\ln c$, giving 0.8813 instead of 0.9133.

To understand parametric survival models, start with the function specifying mortality. A life table records survival at selected ages; the relationships between the two follow by evaluating the same survival probabilities at those ages.

HIGH-FREQUENCY: The relationships between the survival function, force of mortality, and density function appear on nearly every FAM exam.

The survival function $S_0(x)$ gives the probability that a newborn survives to at least age $x$.

If $X$ is the age-at-death random variable:

$$S_0(x) = \Pr(X > x) = 1 - F_0(x)$$

Properties: $S_0(0) = 1$, $S_0(\omega) = 0$, and $S_0(x)$ is non-increasing. Here $\omega$ is the limiting age.

Common mistakes

• Confusing the survival function argument. $S_0(x)$ conditions on birth. ${}_{t}p_x$ conditions on survival to age $x$. Writing ${}_{10}p_{50} = S_0(60)$ is wrong, the correct formula is $S_0(60)/S_0(50)$. Trap: an answer choice that equals $S_0(60)$ alone.
• Forgetting the negative sign in the force of mortality. $\mu_x = -S_0'(x)/S_0(x)$, not $S_0'(x)/S_0(x)$. A negative force of mortality is always wrong.
• Misapplying the Gompertz integral. The integral of $Bc^{x+s}$ from 0 to $t$ is $Bc^x(c^t - 1)/\ln c$, not $Bc^x \cdot t$. Treating $c^{x+s}$ as constant gives a linear approximation. Trap: a value close to correct but systematically off for large $t$.

Bottom line

• Survival function $S_0(x) = \Pr(X > x)$, unconditional from birth; $S_0(0)=1$, $S_0(\omega)=0$
• Force of mortality $\mu_x = f_0(x)/S_0(x) = -\frac{d}{dx}\ln S_0(x)$, the master function that determines the entire model
• Recovery: $S_0(x) = \exp(-\int_0^x \mu_t\,dt)$, exponentiate the negative integral to get survival
• Conditional survival: ${}_{t}p_x = S_0(x+t)/S_0(x) = \exp(-\int_0^t \mu_{x+s}\,ds)$, conditional on reaching age $x$

Exam shortcut

When you see a Gompertz or Makeham problem, write the integral formula immediately: $Bc^x(c^t-1)/\ln c$. Half the work on these problems is evaluating this integral. For life table problems, draw a quick timeline showing which $l$ values sit at which ages, this prevents dividing by the wrong denominator. Recovery rule: "Exponentiate the negative integral of the force to get survival." This one sentence connects $\mu$ to $S$.

The full lesson (about 3,340 words, 22 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 8a

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