Weibull's Law of Mortality : Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3L, authored by Mr. Stolyarov. This is Section 25 of the Study Guide. See an index of all sections by following the link in this paragraph.

Weibull's law of mortality corresponds to the following survival distribution.

s(x) = exp[-uxⁿ⁺¹]

The force of mortality under Weibull's law is as follows.

µ_x = kxⁿ

Under Weibull's law, k and n are given constants, and u is defined as u = k/(n + 1).

The restrictions for using Weibull's law are as follows:

k > 0, n > 0, x ≥ 0.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. p. 78.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L25-1. The lives of rabbit-eared basilisks follow Weibull's law with k = 0.2 and n = 0.4. Find µ₁₈ for rabbit-eared basilisks.

Solution S3L25-1. We use the formula µ_x = kxⁿ. Thus, µ₁₈ = 0.2*18^0.4 = µ₁₈ = 0.6355343046.

Problem S3L25-2. The lives of rabbit-eared basilisks follow Weibull's law with k = 0.2 and n = 0.4. Find s(22) for rabbit-eared basilisks. The answer will be very small!

Solution S3L25-2. We use the formula s(x) = exp[-uxⁿ⁺¹]. We find u = k/(n+1) = 0.2/1.4 = 1/7.

Thus, s(22) = exp[-(1/7)22^1.4] = s(22) = about 0.00001996248653.

Problem S3L25-3. The lives of wild rainbow-spotted anchovies follow Weibull's law with k = 0.11. You know that µ₃ = 0.25 for wild rainbow-spotted anchovies. Find s(10) for wild rainbow-spotted anchovies.

Solution S3L25-3. First, we need to find n by using the formula µ_x = kxⁿ with x = 3.

Hence, µ₃ = 0.11*3ⁿ. Thus, 0.25 = 0.11*3ⁿ and 3ⁿ = 25/11, so n*ln(3) = ln(25/11) and n = ln(25/11)/ln(3) = n = 0.7472887028.

Now we find u = k/(n+1) = 0.11/1.7472887028 = u = 0.0629546793.

Now we use the formula s(x) = exp[-uxⁿ⁺¹] for x = 10.

s(10) = exp[-0.0629546793*10^1.7472887028] = s(10) = about 0.03072459

Problem S3L25-4. The lives of four-legged fish follow Weibull's law with n = 1 and s(20) = 0.14. Find k for four-legged fish.

Solution S3L25-4. We use the formula s(x) = exp[-uxⁿ⁺¹] to recognize that