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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models

For a life aged x with survival function S_x(t) = P(T_x > t), where T_x is the future lifetime, which expression gives the complete expectation of life e°_x, assuming the usual limiting conditions?

The complete expectation of life is the integral of the survival function S_x(t) from 0 to infinity. This follows from integrating E[T_x] by parts. The discrete sum of S_x(t) gives the curtate expectation instead.

  1. AThe integral from 0 to infinity of t·S_x(t) dt
  2. BThe integral from 0 to infinity of S_x(t) dtCorrect
  3. CThe sum over t = 1, 2, ... of S_x(t)
  4. DThe integral from 0 to infinity of μ_{x+t} dt
  5. The integral from 0 to infinity of S_x(t)² dt

Explanation

Since E[T_x] = ∫ t f_x(t) dt, integration by parts with t·S_x(t) tending to 0 gives ∫ S_x(t) dt. The sum of S_x(t) over integers gives the curtate expectation, not the complete one. Option 0 wrongly weights S by t.

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