Curate-Future-Lifetime: Practice Problems and Solutions

The curate-future-lifetime of life (x) is the number of future years that life (x) completes before death. The curate-future-lifetime of (x) is represented by the random variable K(x).

The probability density function of K(x) can be expressed in the following ways:

Pr[K(x) = k] = Pr[k ≤ T(x) < k + 1]

Pr[K(x) = k] = _kp_x - _k+1p_x

Pr[K(x) = k] = _kp_x*q_x+k

Pr[K(x) = k] = _k│q_x

for all nonnegative integer values of k.

The cumulative distribution function of K(x) can be expressed as follows:
Pr[K(x) ≤ k] = _k+1q_x for all nonnegative integer values of k.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1986. First Edition. Society of Actuaries: Itasca, Illinois. p. 48.

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

Problem S3L3-1. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. Find the probability that the curate-future-lifetime of a triceratops life (34) is 7 years.

Solution S3L3-1. We want to find Pr[K(34) = 7].

We use the formula Pr[K(x) = k] = _k│q_x. In this case, _7│q₃₄ = (s(34 + 7) - s(34 + 7 + 1))/s(34) =

(s(41) - s(42))/s(34) = (e^-0.34*41 - e^-0.34*42)/(e^-0.34*34) = Pr[K(34) = 7] = 0.0266758231

Problem S3L3-2. Every year, 0.26 of the lemming population senselessly perishes (i.e., dies), and every lemming has the same likelihood of senselessly perishing. Find the probability that the curate-future-lifetime of a lemming currently aged 3 is 2 years.

Solution S3L3-2. We use the formula Pr[K(x) = k] = _kp_x - _k+1p_x. We want to find Pr[K(3) = 2]. Thus, we need to find ₂p₃ and ₃p₃. The probability of a lemming of any age surviving two more years is

(1 - 0.26)² = 0.5476 so ₂p₃ = 0.5476. The probability of a lemming of any age surviving three more years is (1 - 0.26)³ = 0.405224 so ₃p₃ = 0.405224. Thus, Pr[K(3) = 2] = ₂p₃ -₃p₃ = 0.5476 - 0.405224 = 0.142376

Problem S3L3-3. The life of a giant pin-striped cockroach has the following survival function associated with it: s(x) = 1 - x/94, for 0 ≤ x ≤ 94 and 0 otherwise. Find the cumulative distribution function of the curate-future-lifetime of life (36). Use k as your only variable.

Solution S3L3-3. The cumulative distribution function of K(36) can be expressed as follows:
Pr[K(36) ≤ k] = _k+1q₃₆. We recall from Section 1 that _tq_x = 1 - s(x +t)/s(x) and so

_k+1q₃₆ = 1 - s(37 + k)/s(36) = 1 - (1 - (37+k)/94)/(1 - 36/94) = 1 - ((94 - 37 - k)/94)/(58/94)

= 1 - (57 - k)/58 = (58 - 57 + k)/58 = Pr[K(36) ≤ k] = (1 + k)/58

Problem S3L3-4. For white elephants, the probability that the curate-future-lifetime of life (123) is 4 years is 0.536. The probability that a white elephant that has survived for 127 years will survive for another year is 0.356. Find the probability that a white elephant that has survived for 123 years will survive to age 127.

Solution S3L3-4. We use the formula Pr[K(x) = k] = _kp_x*q_x+k. We are given that Pr[K(123) = 4] is 0.536 and that p₁₂₇ is 0.356. We can quickly find q₁₂₇ = 1 - 0.356 = 0.644. Our goal is to find ₄p₁₂₃. By the formula above, ₄p₁₂₃ = Pr[K(123) = 4]/q₁₂₇ = 0.536/0.644 = ₄p₁₂₃ = 0.8322981366.

Problem S3L3-5. Three-headed donkeys always survive until age 1. Thereafter, the survival function for the life of a three-headed donkey is s(x) = 1/x for all x > 1. Eight-tailed oxen also always survive until age 1. Thereafter, the survival function for the life of an eight-tailed ox is 1/x² for all x > 1. Find the probability that the curate-future-lifetime of a three-headed donkey life (3) is 5 years and that the curate-future-lifetime of an eight-tailed ox life (3) is 5 years.

Solution S3L3-5.

For three-headed donkeys, Pr[K(3) = 5] = _5│q₃ = (s(8) - s(9))/s(3) = (1/8 - 1/9)/(1/3) = 1/24

For eight-tailed oxen, Pr[K(3) = 5] = _5│q₃ = (s(8) - s(9))/s(3) = (1/8² - 1/9²)/(1/3²) = 17/576

To get the probability that both of these curate-future-lifetimes are 5 years, we multiply the individual probabilities and obtain our desired answer: (1/24)(17/576) = 17/13824 = about 0.0012297454.

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