IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
Under a uniform distribution of deaths between integer ages, which relationship between the complete expectation of life e°_x and the curtate expectation of life e_x holds?
Under uniform distribution of deaths, e°_x = e_x + 0.5. The future lifetime equals the curtate lifetime plus a fractional part that averages half a year, so the complete expectation exceeds the curtate by 0.5.
- Ae°_x = e_x
- Be°_x = e_x + 1
- Ce°_x = e_x + 0.5Correct
- De°_x = e_x − 0.5
- e°_x = 0.5 e_x
Explanation
Write T = K + S, where K is the curtate future lifetime and S the fractional part. Under UDD, S is uniform on (0,1) and independent of K, so E[S] = 0.5. Hence e°_x = e_x + 0.5. Option 3 reverses the sign.
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