IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
Which of the following statements about the force of mortality μ(x) and the survival function is correct?
The integral of the force of mortality to infinity must diverge. Since S(x) is the exponential of minus the cumulative hazard and must approach zero, the cumulative hazard must grow without bound. The force is a rate, can exceed 1 and need not increase.
- Aμ(x) must lie between 0 and 1
- Bμ(x) is a probability of death within one year of age x
- CThe integral of μ(x) from 0 to infinity must be finite
- DThe integral of μ(x) from 0 to infinity must diverge to infinityCorrect
- μ(x) must be non-decreasing in x
Explanation
S(x) = exp(−∫μ), and S must tend to 0 as x grows, so the cumulative hazard must diverge. μ is a rate, not a probability, and can exceed 1. Hazard rates need not be monotone, for example in infant mortality.
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