Question

(a) The survival function for a newborn is given by a straight line until the maximum achievable age M. What is the maximum age M if the life expectancy is 60 years? (b) Using your survival function from part (a), find the survival function for a 40 year old. Write your answer in terms of t, where t is the number of years beyond age 40 (so t=0 for a person 40 years old).

(c) A company is hiring 40 year old people who will stay with the company until age 65 , or until death, whichever comes first. Using your answer to part (b), what is the expected time that such a person is with the company?

Answer 1

The maximum age M for a newborn is 55 years when the life expectancy is 60 years. The survival function for a 40 year old is given by S(t) = 1 - t/55. The expected time that a 40 year old person is with the company is approximately 50.45 years.

Step-by-step explanation:

The questions can be answered as -

(a)

Given that the life expectancy is 60 years, we can find the maximum age M by subtracting the life expectancy from the average maximum human lifespan. So, M = 115 - 60 = 55 years.

(b)

To find the survival function for a 40 year old, we need to subtract 40 from the maximum age M. So, the survival function for a 40 year old is given by S(t) = 1 - t/55, where t is the number of years beyond age 40.

(c)

To find the expected time that a 40 year old person is with the company, we need to integrate the survival function from age 40 to age 65. So, the expected time is ∫4065 (1 - t/55) dt = 65 - (40/55)(65-40) = 65 - (40/55)(25) = 65 - (800/55) = 65 - 14.55 ≈ 50.45 years.

Answer 2

(a) The maximum achievable age M is 120 years.

(b) The survival function for a 40-year-old is: S(t) = 1 - (t/80), where t is the number of years beyond age 40.

(c) The expected time spent with the company is 25 years.

Step-by-step explanation:

Part (a):

A straight line survival function implies a constant decrease in survival probability with age.

Life expectancy is the average number of years a person lives. In this case, 60 years.

Since the survival function decreases linearly until age M, the area under the curve from 0 to M represents the total life expectancy.

This area can be calculated as the product of the average lifespan (60 years) and the maximum age M: 60M = (M/2) * M.

Solving for M: 60 = M/2 => M = 120 years.

Part (b):

At age 40, the person has lived for 40 years, so the survival probability S(0) is equal to S(40) in the initial linear function.

As the function intersects the x-axis at M=120, we can find the slope as: (S(0) - 0) / (0 - M) = (S(40) - 0) / (40 - 120)

Substituting S(40) with 1 and M with 120: 1 / (-80) = S(40) - 0 => S(40) = 1 - 1/80 = 79/80.

Since the survival function decreases linearly, S(t) for any age t beyond 40 is: S(t) = S(40) - (t/80) = 79/80 - (t/80) = 1 - (t/80)

Part (c):

The expected time spent with the company is the average of the remaining lifespans for the 40-year-old employees.

This average can be calculated using the weighted average of the survival function across the relevant age range (40-65):

Expected time = ∫_0^(25) (t + 40) * S(t) dt

Substituting the survival function from part (b): Expected time = ∫_0^(25) (t + 40) * (1 - (t/80)) dt

Solving this integral (e.g., using numerical methods) gives an expected working time of 25 years.