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Max, min, and an exercise

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  • Max, min, and an exercise

    Hello all,

    I want to have a better understanding of max and min.
    So far, I see min as the subset and max as something that is overlaying.
    I'm not absolutely sure about that.

    On SOA 153, #46.
    A device that continuously measures and records seismic activity is placed in a remote
    region. The time, T, to failure of this device is exponentially distributed with mean
    3 years. Since the device will not be monitored during its first two years of service, the
    time to discovery of its failure is X = max(T, 2) .
    Determine E[X].

    Solution goes as:
    I know the pdf of expo(1/3) and how to generally get EX.

    But they add two integrals:

    int(2f(x) from 0, 2) + int(xf(x) from 2,inf).
    f(x) = pdf of expo(1/3)

    Why is int(2f(x) from 0, 2) included? Where does the multiplier 2 come from?

  • #2
    I wouldn't get too confused in what Min and Max mean here in these kinds of problems...

    How do you calculate E[x] normally?


    In this case though... from 0 to 2 years we won't monitor the device or machine or what ever the problem asks. So if it SHOULD fail sometime in that boundary.... we would not know when and would chalk it up to as soon as we started monitoring it (the two year mark) so instead of x*f(x) we'd have 2*f(x)... but had it not failed in that time period and was still working when we begin monitoring we treat it as normal from 2 to infinity that is to say x*f(x)

    Does that clarify things better? I find it better to just describe the problem to yourself and not to get too lost on the Max and Min here.


    • #3
      That explanation works but if you want to think about how max function is defined ... and you should know this: max(T,2) = 2 if T < 2 and T elsewhere. Therefore when T is between 0 and 2, you integrate 2*pdf dt and when T is greater than 2 you integrate t*pdf dt. Luckily this is a cont. pdf and you don't need to worry about any singular points (such as T = 2) but you should know how to handle that point if it was a discrete distribution.