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Math Help - Find the pdf

  1. #1
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    Find the pdf

    Let X be a Uniform(0,1) random variable. Find the pdf of Y = (-1/lamda)*ln(X) , where lamda is a given positive number.

    I don't think this is very difficult, but I'm not really sure how to go about it when X and Y are together like that. If anyone could point me in the right direction I'd really appreciate it!
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  2. #2
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    Hint -
    1. Try plotting y as f(x)? What is the range it takes?
    2. Can you find the cdf for y? i.e. Prob (y<y0)
    3. Differentiate the cdf to get pdf

    This should not be too tough
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  3. #3
    MHF Contributor matheagle's Avatar
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    Use Calc 1

    f_Y(y)=f_X(x)\left|{dx\over dy}\right|
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  4. #4
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    Quote Originally Posted by jlt1209 View Post
    Let X be a Uniform(0,1) random variable. Find the pdf of Y = (-1/lamda)*ln(X) , where lamda is a given positive number.

    I don't think this is very difficult, but I'm not really sure how to go about it when X and Y are together like that. If anyone could point me in the right direction I'd really appreciate it!
    Quote Originally Posted by matheagle View Post
    Use Calc 1

    f_Y(y)=f_X(x)\left|{dx\over dy}\right|
    Alternatively, since X ~ U(0, 1) => 1 - X ~ U(0, 1) it follows immediately from the probability integral transform theorem that Y is an exponential random variable with pdf \lambda e^{- \lambda y}.
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