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Math Help - Another exponential dist.

  1. #1
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    Another exponential dist.

    The lifetime of a product is Y = 5X^0.7 , where X has an exponential dist. with mean = 1

    Find the distribution function and pdf of Y

    I'm not even sure where to start here...
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  2. #2
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    The lifetime of a product is Y = 5X^0.7 , where X has an exponential dist. with mean = 1


    So the cdf,  F(x) = \mathbf{P}(Y\leq x) = \mathbf{P}(5X^{0.7}\leq x) = \mathbf{P}(X\leq (x/5)^{1/.7})

    The last expression is in the form of the CDF of an exponential. ( \mathbf{P}(X\leq x_o) = 1 - exp(-x/\mu) ) And so

     = 1-exp(-(x/5)^{1/.7})

    To get the pdf, just take the derivative of the CDF....
    Last edited by meymathis; July 10th 2008 at 02:58 PM. Reason: typo
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  3. #3
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    I did that, but it's not the correct answer.
    Unless I'm doing something wrong
    here's the answer:

    [10/7*y^(3/7)*e^(-y/5)^10/7] / 5^(10/7)

    Looks like your answer is pretty close, so I assume it's me doing something wrong. Thanks for the help
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  4. #4
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    derivative of the CDF:
     \frac{d}{dx}\left( 1-\exp(-(x/5)^{1/.7}) \right)<br />

    derivative of the sum is sum of the derivatives

    derivative of constant is 0

    1/.7 = 10/7

    = 0 +\frac{d}{dx}\left(-\exp(-(x/5)^{10/7}) \right)<br />

    Now \frac{d}{dx}\exp(u) = exp(u)\frac{du}{dx} (derivative of exp with chain rule). So

    = -\exp(-(x/5)^{10/7}) \frac{d}{dx} \left(-(x/5)^{10/7}\right)

    Then

    = -\exp(-(x/5)^{10/7}) \frac{-10}{7}(x/5)^{10/7-1}\frac{d}{dx}(x/5)

    = \frac{10}{7} exp(-(x/5)^{10/7}) x^{3/7}\frac{1}{5^{3/7}5}

    = \frac{10}{7} \frac{1}{5^{10/7}}exp(-(x/5)^{10/7}) x^{3/7}

    which is what you had
    [10/7*y^(3/7)*e^(-y/5)^10/7] / 5^(10/7)
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  5. #5
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    Yea, I actually figured it out like an hour ago. Been too many years since calculus; thanks for all your help!
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  6. #6
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    Quote Originally Posted by amor_vincit_omnia View Post
    The lifetime of a product is Y = 5X^0.7 , where X has an exponential dist. with mean = 1

    Find the distribution function and pdf of Y

    I'm not even sure where to start here...
    Popular question. Asked and answered (in different ways) in the following threads:

    http://www.mathhelpforum.com/math-he...ion-p-d-f.html

    http://www.mathhelpforum.com/math-he...ction-f-x.html

    http://www.mathhelpforum.com/math-he...-lifetime.html
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