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Soa #108

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  • Soa #108

    Question:

    A device containing two key components fails when, and only when, both components fail. The lifetimes, T1 and T2, of these components are independent with common density function f (t) = e−t, t > 0 . The cost, X, of operating the device until failure is 2T1 + T2 . Which of the following is the density function of X for x > 0 ?

    In the solution, the beginning steps are:

    P(X<x)=P(2T1+T2<x)
    =integral from 0 to x of [integral from 0 to 0.5(x-T2) of (e^-t1)(e^-t2)dt1] dt2

    I don't understand where the limits for this integral came from. Specifically the dt2 one, from 0 to x. Could someone please clarify where it came from?

  • #2
    This thread discusses that problem (and 109)
    http://www.actuary.com/actuarial-dis...hlight=soa+109

    From what I understand, drawing a picture of the situation would definitely help, pretending that x is a constant, so then you have a picture or a triangle with limits 0<t1<x-t2 and 0<t2<x or something like that.

    And if you move onto 109 and have questions why 108's limit goes to x and why 109's goes to infinity drawing a picture for 109 will also help (basically that line has a positive slope and goes to infinity, which is why the limit is infinity)

    Hope that helps! And yeah, check out that thread because you might understand it better.

    Comment


    • #3
      I drew them both out and I understand their limits now. Thanks for the quick response!

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