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

For a life aged x, the force of mortality is mu_(x+t) = -d/dt ln(tp_x). Which expression gives the probability density function of T_x at time t?

The density of T_x at t is tp_x multiplied by mu_(x+t), the probability of surviving to t times the force of mortality at age x+t. This follows from differentiating the survival function, whose derivative equals minus tp_x times the force of mortality.

  1. Atp_x * mu_(x+t)Correct
  2. Btp_x / mu_(x+t)
  3. C(1 - tp_x) * mu_(x+t)
  4. Dmu_(x+t) / tp_x
  5. tp_x * mu_x

Explanation

The density is f(t) = -d/dt tp_x. Since d/dt tp_x = -tp_x * mu_(x+t), the density equals tp_x * mu_(x+t). Option 3 uses the distribution function in place of the survival function, and option 5 uses the force at the wrong age.

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