The Uniform Distribution of Deaths Assumption for Fractional Ages: Practice Problems and Solutions: Part 2

We define the following random variables:

T = time until death

K = curate-future-lifetime

S = the fractional part of a year lived in the year of death

The following hold irrespective of the assumptions we make for fractional ages.

T = K + S

Pr[k < T ≤ k + s] = _k│sq_x = _kp_x*_sq_x+k

Then, under the assumption of the uniform distribution of deaths for fractional ages, the following relationships hold:

Pr[k < T ≤ k + s] = s*_k│q_x = s*_kp_x*q_x+k

ė_x = e_x + ½

Var[T] = Var[K] + 1/12

Meaning of variables:

ė_x= complete-expectation-of-life for a life currently at age x.

e_x= expected value of the curate-future-lifetime for a life currently at age x.

The variables K and S are independent under the uniform distribution of deaths assumption.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1986. First Edition. Society of Actuaries: Itasca, Illinois. pp. 70-71.

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

Problem S3L16-1. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. For a 4-year-old triceratops, find Pr[12 < T(4) ≤ 12.67] without assuming a uniform distribution for fractional ages or making any other special assumptions.

Solution S3L16-1. We use the formula Pr[k < T ≤ k + s] = _k│sq_x.

Thus, Pr[12 < T(4) ≤ 12.67] = _12│0.67q₄ = (s(4 + 12) - s(4 + 12 + 0.67))/s(4) = (s(16) - s(16.67))/s(4) =

(e^-0.34*16 - e^-0.34*16.67)/(e^-0.34*4) = Pr[12 < T(4) ≤ 12.67] = about 0.0034443297.

Problem S3L16-2. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. For a 4-year-old triceratops, find Pr[12 < T(4) ≤ 12.67] by assuming a uniform distribution of deaths for fractional ages.

Solution S3L16-2. We use the formula Pr[k < T ≤ k + s] = s*_k│q_x. Thus, Pr[12 < T(4) ≤ 12.67] =

0.67_12│q₄ = 0.67(e^-0.34*16 - e^-0.34*17)/(e^-0.34*4) = Pr[12 < T(4) ≤ 12.67] = about 0.0032650664.

Problem S3L16-3. For a certain population of newborn rabid rabbits, the expected value of the curate-future-lifetime is 5.56 years. Using the assumption of the uniform distribution of deaths for fractional ages, find the complete-expectation-of-life for newborn rabid rabbits.

Solution S3L16-3. The expected value of the curate-future-lifetime is defined as E[K] = e_x. The complete-expectation-of-life is ė_x. We want to find ė₀.We thus use the formula ė_x = e_x + ½.

ė₀ = e₀ + ½ = 5.56 + 0.5 = 6.16 years.

Problem S3L16-4. The variance of the total future lifetime of 7-year-old giant dragonflies is 4. Find the variance of the curate-future-lifetime of 7-year-old giant dragonflies.

Solution S3L16-4. We use the formula Var[T] = Var[K] + 1/12, where we want to find Var[K] = Var[T] - 1/12 = 4 - 1/12 = 47/12 = about 3.916666666667

Problem S3L16-5. Augustus the Actuary makes a bet with Gilgamesh the Gambler that it is possible to reasonably analyze the populations of √(3)-winged hippopotami by assuming a uniform distribution of deaths for fractional ages. Augustus needs to use the assumption to figure out the probability that a newborn √(3)-winged hippopotamus will die sometime between age 5 and age 5.33. Augustus will win the bet if his result using the assumption is within 0.003 of the true result. It is known that the survival function for √(3)-winged hippopotami is s(x) = exp[-0.00003(5^x - 1)]. Who will win the bet? Justify your answer.

Solution S3L16-5. We need to find Pr[5 < T(0) ≤ 5.33] in two ways.

First, without any assumptions,

Pr[5 < T(0) ≤ 5.33] = _5│0.33q₀ = (s(5) - s(5.33))/s(0) = (s(5) - s(5.33)) =

exp[-0.00003(5⁵ - 1)] - exp[-0.00003(5^5.33 - 1)] = 0.0996302456. We will call this value A.

Now, assuming a uniform distribution of deaths for fractional ages,

Pr[5 < T(0) ≤ 5.33] = 0.33*_5│q₀ = 0.33(s(5) - s(6))/s(0) = 0.33(s(5) - s(6)) =

0.33(exp[-0.00003(5⁵ - 1)] - exp[-0.00003(5⁶ - 1)]) = 0.0939625149. We will call this value B.

Now we need to find A - B = 0.0996302456 - 0.0939625149 = 0.0056677309 > 0.003, so Gilgamesh wins the bet.

See other sections of The Actuary's Free Study Guide for Exam 3L.