IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
In the standard survival model notation, T_x denotes the future lifetime of a life aged x. Which of the following correctly expresses the survival function of T_x in terms of the survival function S_0 of the age-at-death random variable X?
The survival function of T_x is S_0(x+t) divided by S_0(x). This is the conditional probability of surviving to age x+t given survival to age x, so the survival function at the older age is divided by the survival function at the current age.
- AP(T_x > t) = S_0(x + t) / S_0(x)Correct
- BP(T_x > t) = S_0(t) / S_0(x)
- CP(T_x > t) = S_0(x + t) - S_0(x)
- DP(T_x > t) = S_0(x) / S_0(x + t)
- P(T_x > t) = S_0(x + t) * S_0(x)
Explanation
T_x is the remaining lifetime given survival to x, so P(T_x > t) = P(X > x+t | X > x) = S_0(x+t)/S_0(x). Option 3 inverts the ratio, and the others do not condition on survival to age x.
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