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Math Help - Exponential distribution.

  1. #1
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    Exponential distribution.

    the time to failure of a machine when used for 1 hour can be modeled by a exponential distribution with Beta value = 50.

    Find the probability that the machine will not fail during 5 consecutive 8 hour shifts.


    In this modified case should i take Beta value as = 50 * 8 * 5 ?

    Please help me.
    Last edited by mr fantastic; May 27th 2010 at 08:15 PM. Reason: Re-titled.
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  2. #2
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    The exponential distribution can be written as {\lambda}=\frac{1}{\beta}

    So, {\lambda}e^{{-\lambda}x} becomes

    \frac{1}{\beta}e^{\frac{-x}{\beta}}

    Beta is called the scale parameter and lambda the rate parameter.

    5-8 hour shifts is 40 hours. The probability it does not break down in this

    time can then be found by integrating over 0 to 40 and subtracting from 1

    1-\frac{1}{50}\int_{0}^{40}e^{\frac{-x}{50}}dx

    Or, if the machine does not break down within 40 hours, then it is sure to breakdown some time after that. Whenever that may be.

    \frac{1}{50}\int_{40}^{\infty}e^{\frac{-x}{50}}dx
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  3. #3
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    if the no of machine's that fails during 5 consecutive 8 hour shifts may be modeled by a Poisson distribution

    what will the value of lambda be

    Please help.
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  4. #4
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    In your beta notation:

    \lambda = \beta^{-1}


    Reasoning:
    If you write your exponential pdf in the form
     f(x)=ae^{-ax}

    Then it models the time to failure of a poisson process with parameter a. Your pdf is in that form with a=1/\beta

    This is why you will often see the exponential distribution written as exp(\lambda). in fact i have never seen exp(\beta) until you asked this question.
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  5. #5
    MHF Contributor matheagle's Avatar
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    I would think is a binomial with n=5 (the shifts)
    Where p is the probability of not failing during an eight hour period, which you can calculate via the exponential distribution.
    But the exponential denisty can be written either way.
    So I don't know where the parameter belongs.
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