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Please explain how the term in square brackets is zero for t = w - x

+1 vote
asked Mar 14, 2017 in BUS 3018F - Models by ChanceTheRapper (650 points)
edited Mar 15, 2017 by ChanceTheRapper

Pg 15 of the ActEd notes (Chapter 7) - the 2nd last line where the bounds of the integral are being evaluated

1 Answer

+3 votes
answered Mar 15, 2017 by Ayesha Camroodien (610 points)
selected Mar 16, 2017 by Rowan
Best answer

The expression in the square bracket [ (t) x (tpx) ] evaluated at t=0 is zero.

And the expression [ (t) x (tpx) ] evaluated at t=(w-x)  is zero for the following reasons:

  • The probability that a life aged x survives for at least (w-x) years is 0
  • w is defined as the maximum age of an individual i.e. it is assumed the individual will not live past the age w (it is the limiting age)
  • (w-x) is the maximum number of years a life could still live from its current age x
  • so if a life has already survived x years, the probability that it survives at least  a further (w-x) years would take the individual past its limiting age (try drawing a timeline to illustrate the concept)
  • in other words, the probability that a life now aged x years old is still alive after (w-x) years is 0
Drawing a timeline can be quite useful when working with survival probabilities!