Parametric Survival Model Calculations

Problem Preview: Given $S_0(x) = (1 - x/100)^{1/2}$, you need $\mathring{e}_{36}$ and $\text{Var}(T_{36})$. The integral $\int_0^{64}(1-t/64)^{1/2}\,dt$ yields 128/3 = 42.667 years, but if you forget the factor of 2 in the second moment formula, you get half the correct variance.

HIGH-FREQUENCY: The distinction between $T_x$ and $K_x$, and the formulas for their expectations, appear on nearly every FAM exam.

Complete future lifetime $T_x$: exact continuous time until death. If a person aged 40 dies at 57.3, then $T_{40} = 17.3$.

Curtate future lifetime $K_x = \lfloor T_x \rfloor$: the number of complete years lived. In the same example, $K_{40} = 17$.

KEY: Circle accent ($\mathring{e}_x$) = continuous/complete (integration). No circle ($e_x$) = discrete/curtate (summation). Under UDD, they differ by exactly 0.5.

$$f_{T_x}(t) = {}_{t}p_x \cdot \mu_{x+t}$$

$$\Pr(K_x = k) = {}_{k}p_x \cdot q_{x+k} = {}_{k\vert}q_x$$

Common mistakes

• Confusing $\mathring{e}_x$ and $e_x$. The circle means continuous (integration). No circle means curtate (summation). Under UDD they differ by 0.5. Trap: an answer choice exactly 0.5 away from correct.
• Starting the curtate sum at $k=0$ instead of $k=1$. Since ${}_{0}p_x = 1$, starting at 0 adds 1 to the correct answer. Trap: a value exactly 1 more than the correct curtate expectation.
• Forgetting the factor of 2 in the second moment. $E[T_x^2] = \int 2t \cdot {}_{t}p_x\,dt$, not $\int t \cdot {}_{t}p_x\,dt$. Omitting the 2 halves the second moment.

Bottom line

• Complete future lifetime $T_x$ is continuous; curtate $K_x = \lfloor T_x \rfloor$ is discrete, and the exam tests both
• $\mathring{e}_x = \int_0^{\omega-x} {}_{t}p_x\,dt$ (complete, integration); $e_x = \sum_{k=1}^{\infty} {}_{k}p_x$ (curtate, sum starts at $k=1$)
• Under UDD: $\mathring{e}_x = e_x + \frac{1}{2}$
• Second moment carries a factor of 2: $E[T_x^2] = \int 2t \cdot {}_{t}p_x\,dt$

Exam shortcut

When a problem gives you a parametric model, immediately identify it. De Moivre and constant force have closed-form answers, write them from memory. For Gompertz and Makeham, expect numerical integration or partial results. "Circle = Continuous, Plain = Integer." The circle on $\mathring{e}_x$ means continuous. No circle on $e_x$ means curtate. "2t in the second moment", visualize the 2 as a superscript that migrated from $T^2$ into the integrand.

The full lesson (about 2,880 words, 19 min read) adds 6 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 8b

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