MATH 403A, fall 98, CHAPTER 3 Section 3.7

Analytical Laws of Mortality

Name  μ(x)  s(x)  Restrictions

De Moivre (w − x)^{−1}  1 − \frac{x}{w}  0 ≤ x < w

Gompertz  Bc^{x}  e^{−m(c^{x}−1)}  B > 0, c > 1, x ≥ 0

Makeham A + Bc^{x} e^{−Ax−m(c^{x}−1)} B > 0, A ≥ −B, c > 1, x ≥ 0

Weibull  kx^{n}  e^{−ux^{n+1}}  k > 0, n > 0, x ≥ 0

where

m = \frac{B}{log c} u = \frac{k}{n + 1}.

Note: De Moivre’s Law states that T (x) has the uniform distribution. Likewise, the constant force analytical law of mortality simply states that T (x) has the exponential distribution. You know a lot of things about these distributions already.

Homework

1. Complete the following table.

de Moivre’s Law Constant force of mortality

tq_{x}

tp_{x}

μ(x + t)

tp_{x}μ(x + t)

{∘ \atop e}

Var[T (x)]

2. A life aged x is subject to deMoivre’s law. Find the probability that the time until death is within one standard deviation of its expected value.

3. You are given that μ(x) = .02 fro x ≥ 0. You are also given that tp_{x} = .5. Find t.

4. The future lifetime random variable for a life aged x has the same variance under two different mortality assumptions: (1) constant force of mortality equal to .06, and (2) de Moivre’s law with w = 100. Find x.

5. Textbook problems: 3.35.