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Which definition(s) of Pareto distribution is used in exams?

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  • Which definition(s) of Pareto distribution is used in exams?

    1. F(x) = 1 - (x0/x)^a, x>x0, or
    2. F(x) = 1 - (x0/(x+x0))^a, x>0 ?

    The first, as well as I understand, is a standard definition for 2-parametric Pareto distribution. Two of the recommended manuals for exam P - Hassett & Stewart, and Wackerly, Mendenhall, & Scheaffer - use the definition.

    The second definition is used by Bean.

    The three other recommended manuals describe 1-parameter Pareto distribution only, using definition 1 with x0=1.

    The expected values calculated according to definitions 1 and 2 differ by x0. For example, in the following mock exam problem

    http://www.math.ilstu.edu/krzysio/4-2-5-KO-Exercise.pdf

    the answers would be $750 and $250 respectively.

    Which of the two definitions is considered standard in actuarial exams?
    Last edited by yurakm; September 7 2005, 12:32 AM.

  • #2
    Unless stated as a "single parameter" Pareto, always assume it is a two parameter Pareto.

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    • #3
      Originally posted by Mathguy36
      Unless stated as a "single parameter" Pareto, always assume it is a two parameter Pareto.
      Thank you. If the problem asks for 2 parameters, it is 2-parameter distribution. However, what to do with two different definitions of 2-parameter Pareto distributions above?

      Let us consider the problem mentioned above:

      http://www.math.ilstu.edu/krzysio/4-2-5-KO-Exercise.pdf

      Two different "right" answers are possible, depending on definition of 2-parameter Pareto distribution. "B" and "D" ($250 and $750). "B" if you used the Bean's book while preparing, or "D" if you read instead Hassett & Steward or many other manuals.

      The Bean's definition can be used to describe insurance payments after deductibles, while the more standard definition describes the loss, including deductibles. In substance this is the same information. Just need to understand what is expected during exam.

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      • #4
        Your #1 is for a single parameter Pareto.
        Your #2 is for a two parameter Pareto. Unless they state "single parameter Pareto" use #2 for F(x). Your original posting is in error. It's been a while since I've sat for P, so I'm not familiar with those textbooks and how they explain the distributions. For a complete listing of distributions (You don't have to memorize them all) download the tables from exam M.

        The link is:
        http://www.soa.org/ccm/cms-service/s...=10959090&g11n


        Hope that helps.

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        • #5
          Thank you very much!

          According the SOA table for Exam M,
          Note: Although there appears to be two parameters <in formula 1>, only a is a true parameter. The value of <x0> must be set in advance.
          So, shifting along the X coordinate makes one-parameter function a two-parameter function. Hard to believe, but useful to know at exam...
          Last edited by yurakm; September 8 2005, 04:01 PM.

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