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Math Help - distribution function F(X)

  1. #1
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    distribution function F(X)

    The lifetime of a product is Y=5X^{0.7} where x has an exponential distribution with mean 1. I need to find the distribution function and pdf of Y.

    For the pdf I got e^{-x}. Can this be simply integrated to get the distribution function F(Y) to equal 1-e^{-x}? I obviously need the Y=5X^{0.7} for something, but I'm not sure how to incorporate it besides subbing x=0.1003393821Y^{10/7}

    Thank you!
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  2. #2
    MHF Contributor matheagle's Avatar
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    calculus one
    f_Y(y)=f_X(x)\biggl|{dx\over dy}\biggr|
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  3. #3
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    I'm sorry. I really need a calculus refresher. So you're saying that the pdf of y is equal to the pdf of x multiplied the derivative of pdf of x? like e^{-x}*(-e^{-x})?
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  4. #4
    MHF Contributor matheagle's Avatar
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    NO....

    the pdf of y is equal to the pdf of x multiplied the derivative of THE CHANGE OF VARIABLES

    read below....
    Dependent variables and change of variables
    in....
    http://en.wikipedia.org/wiki/Probabi...nsity_function
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  5. #5
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    thank you for the link. I found this under change of variables. I'm sure I am mistaken, but I though the following means the derivative of the pdf of y equals the derivative of the pdf of x: That would then mean they are equal and both e^{-x}, which can't be right. I apologize for being so confused.
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  6. #6
    MHF Contributor matheagle's Avatar
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    That's the same as what I wrote.

    Use this .... f_Y(y)=f_X(x)\biggl|{dx\over dy}\biggr|

    Plug in your y for x in f_X(x)=e^{-x}, but solve for x first.

    Next obtain the derivative {dx\over dy} in terms of y.

    Multiply the two and it's over.
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  7. #7
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    solving for x in y=5x^{0.7}, i got x=0.100339y^{10/7}. Plugging that in for the x in e^{-x} and taking the derivative, I got -0.14334y^{3/7}e^{-0.10034y^{10/7}}. Multiplying this by the first equation, I ended up with the big messy equation -0.14334y^{3/7}(e^{-0.10034y^{10/7}})^{2}
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  8. #8
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    so would be my distribution function F(y)?

    After differentiating this messy equation I have the p.d.f. f(y)= -\frac{0.0614(e^{(-0.10034y^{10/7})^{2}}}{y^{4/7}}+0.04109y^{6/7}(e^{(0.10034y^{10/7})^{2}}.

    This all looks too messy to be correct. Can somebody help point me in the right direction? This is confusing me so much. The more I read, the more confused I am getting. Thank you so much for your help!

    (cont)
    I obviously should have used x=(\frac{y}{5})^{\frac{10}{7}}

    f(x)=e^{-(\frac{y}{5})^{\frac{10}{7}}}
    \frac{dx}{dy}=\frac{-2}{35}*5^{\frac{4}{7}}*y^{\frac{3}{7}}*e^{\frac{-1}{25}*5^{\frac{4}{7}}*y^{\frac{10}{7}}}
    multiplying the two together, I got \frac{-2}{35}*5^{\frac{4}{7}}*y^{\frac{3}{7}}(e^{\frac{-1}{25}*5^{\frac{5}{7}}*y^{\frac{10}{7}}})^{2} which looks a little better but still doesn't seem right. Any help would be extremely appreciated.
    Last edited by laser; April 27th 2009 at 12:02 PM.
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  9. #9
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    Quote Originally Posted by laser View Post
    The lifetime of a product is Y=5X^{0.7} where x has an exponential distribution with mean 1. I need to find the distribution function and pdf of Y.

    For the pdf I got e^{-x}. Can this be simply integrated to get the distribution function F(Y) to equal 1-e^{-x}? I obviously need the Y=5X^{0.7} for something, but I'm not sure how to incorporate it besides subbing x=0.1003393821Y^{10/7}

    Thank you!
    Popular question. Asked and answered (in different ways) in the following threads:

    http://www.mathhelpforum.com/math-he...tial-dist.html

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

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

    There's more than enough help been given to answer this question.

    Thread closed.
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