IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
Let T_x have survival function tp_x = (1 - t/60)^2 for 0 <= t <= 60, where x is a given age. What is the complete expectation of life, the mean of T_x?
The mean future lifetime is 20 years. It equals the integral of the survival function from 0 to 60, which is 60 times the integral of u squared from 0 to 1, giving 60 divided by 3. A linear survival function would wrongly give 30.
- A20Correct
- B30
- C15
- D40
- 25
Explanation
Mean = integral from 0 to 60 of (1 - t/60)^2 dt. Substituting u = 1 - t/60 gives 60 * integral of u^2 du over 0 to 1 = 60/3 = 20. Check: it is a Beta-type distribution with mean 60/3. The value 30 results from using a linear survival function.
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