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  • Complete Expectation of Life assuming UDD

    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.

  • #2
    I don't recall ever having seen one, but I have also done every life con question I could get my hands on, and I never was in a situation in which I would need a formula like that.

    Originally posted by willborg35 View Post
    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.

    Comment


    • #3
      Originally posted by willborg35 View Post
      Under UDD it is assumed exactly that:

      Complete expectation of Life = Kurate Expectation of Life + 1/2

      Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?
      Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
      There is an approximation: you add (1/2)*(n_q_x) instead of 1/2.

      ctperng

      Comment


      • #4
        Go

        Originally posted by willborg35 View Post
        Under UDD it is assumed exactly that:

        Complete expectation of Life = Kurate Expectation of Life + 1/2

        Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

        Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
        This is easy to figure out

        Complete: e_{x:n} = integral(t_p_x dt from 0 to n) = integral(1 - t/(w-x) dt from 0 to n) = n - n^2/(2(w-x)) = n[1 - n/[2(w-x)]]

        Curtate (not Kurate): e_{x:n} = sum(t_p_x from t=1 to n - 1) = sum(1 - k/(w-x) from 1 to n-1) = (n-1) - n(n-1)/[2(w-x)] = (n-1)[1 - n/[2(w-x)]] = n[1 - n/[2(w-x)]] - [1 - n/[2(w-x)]]

        So as you can see Curtate = Complete - [1 - n/[2(w-x)]]

        Obviously if the sum goes to "infinity" i.e. in this case to w - x, we let n = w-x and we get:

        Curtate = Complete- [1 - (w-x)/[2(w-x)]] = Complete - [1 - 1/2] = Complete - 1/2

        Hopefully my algebra is right.
        ________
        jugallette
        Last edited by NoMoreExams; January 20 2011, 06:28 PM.

        Comment


        • #5
          Originally posted by NoMoreExams View Post
          This is easy to figure out

          Complete: e_{x:n} = integral(t_p_x dt from 0 to n) = integral(1 - t/(w-x) dt from 0 to n) = n - n^2/(2(w-x)) = n[1 - n/[2(w-x)]]

          Curtate (not Kurate): e_{x:n} = sum(t_p_x from t=1 to n - 1) = sum(1 - k/(w-x) from 1 to n-1) = (n-1) - n(n-1)/[2(w-x)] = (n-1)[1 - n/[2(w-x)]] = n[1 - n/[2(w-x)]] - [1 - n/[2(w-x)]]

          So as you can see Curtate = Complete - [1 - n/[2(w-x)]]

          Obviously if the sum goes to "infinity" i.e. in this case to w - x, we let n = w-x and we get:

          Curtate = Complete- [1 - (w-x)/[2(w-x)]] = Complete - [1 - 1/2] = Complete - 1/2

          Hopefully my algebra is right.
          I got something else, when you do the n-year curtate temporary life expectancy you get

          summation of (tPx)

          which is 1 - k/(w-x) from 1 to n (NOT n-1)

          solving the summation would give you

          n - (summation of k from 1 to n)/(w-x) this would give you

          n - n(n+1)/2(w-x) which is n - n^2/2(w-x) - n/2(w-x)

          so that is continuous e_x:n = curtate e_x:n + n/2(w-x) which would only be 1/2 when n = w-x


          which is what ctperng was saying
          Last edited by zmkramer; April 23 2008, 04:04 PM.

          Comment


          • #6
            Originally posted by zmkramer View Post
            I got something else, when you do the n-year curtate temporary life expectancy you get

            summation of (tPx)

            which is 1 - k/(w-x) from 1 to n (NOT n-1)

            solving the summation would give you

            n - (summation of k from 1 to n)/(w-x) this would give you

            n - n(n+1)/2(w-x) which is n - n^2/2(w-x) - n/2(w-x)

            so that is continuous e_x:n = curtate e_x:n + n/2(w-x) which would only be 1/2 when n = w-x


            which is what ctperng was saying

            booyah
            I agree, my summation should go to n not n - 1, however our results agree, when you plug in n = w - x, you get cont = curtate + 1/2, I wrote down curtate = cont - 1/2.

            Comment


            • #7
              Originally posted by willborg35 View Post
              Under UDD it is assumed exactly that:

              Complete expectation of Life = Kurate Expectation of Life + 1/2

              Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

              Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
              Actually, it's ex:n + .5(1-nPx).

              Comment

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