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Math Help - Want to confirm use of gamma distribution

  1. #1
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    Want to confirm use of gamma distribution

    I want to confirm that after finding the convolution of the independent variables I need to use the gamma distribution to solve the following problem:

    A Mars probe has two batteries. Once a battery is activated, its future lifetime is exponential with mean 1 year. The first battery is activated when the probe lands on Mars. The second battery is activated when the first fails. Battery lifetimes after activation are independent. The probe transmits data until both batteries have failed. Calculate the probability that the probe is transmitting data three years after landing.
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  2. #2
    MHF Contributor matheagle's Avatar
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    the sum of two independent exponentials is a gamma
    You want P(X+Y>3).
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  3. #3
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    So, if there were more than two independent exponentials, then I would use the gamma distribution formula?
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  4. #4
    MHF Contributor matheagle's Avatar
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    Using MGFs we can easily prove that

    If X\sim \Gamma(\alpha_1,\beta) and Y\sim \Gamma(\alpha_2,\beta)

    (where they are independent), then

    X+Y\sim \Gamma(\alpha_1+\alpha_2,\beta).


    This is very useful for adding Chi-Squares, which is important in Design of Experiments, where
    we use Cochran's Theorem repeatedly.
    http://en.wikipedia.org/wiki/Cochran's_theorem
    Last edited by matheagle; December 6th 2009 at 11:55 PM.
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