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

For a life aged x with future lifetime random variable T_x, the complete expectation of life is e°_x. Which expression correctly gives it, where S_x(t) = t p_x?

The complete expectation of life e°_x equals the integral from zero to infinity of the survival probability t p_x. This follows because the mean of a non-negative variable is the integral of its survival function. The integer sum of t p_x gives the curtate expectation instead.

  1. AThe sum over integer t≥1 of t p_x
  2. BThe integral from 0 to infinity of t p_x dtCorrect
  3. CThe integral from 0 to infinity of t · t p_x dt
  4. DThe integral from 0 to infinity of μ_{x+t} dt
  5. The sum over integer t≥0 of t q_x

Explanation

E[T_x] equals the integral of the survival function of T_x, which is t p_x, from 0 to infinity. The sum over integers is the curtate expectation e_x, not the complete expectation. Option C is not a valid form of the mean, since t multiplied by the survival function does not equal E[T_x].

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