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Math Help - Probability Density Function of a function

  1. #1
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    Probability Density Function of a function

    Dear All,

    let us say we have PDF(x) i.e. probability density function of variable x. How to find probability density function for f(x), PDF(f(x))?
    Particularly

    PDF(x) is



    and f(x)=x2

    Thanks in advance.
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  2. #2
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    Re: Probability Density Function of a function

    Sorry expression for the PDF(x) did not appear it is PDF for Gaussian distribution.
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  3. #3
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    Re: Probability Density Function of a function

    I think that

    PDF_f(x)(x)=const*PDF(inversf(x))

    if PDF(x) is Gaussian and f(x)=x^2;
    then PDF_x^2(x)=const*PDF(sqrt(x)) so the x should be replaced by sqrt(x) in Gaussian PDF in case of 0 mean. In case non 0 man x-mean should be replaced by sqrt(x-mean).
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  4. #4
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    Re: Probability Density Function of a function

    You are partially right. F_{f(X)}(y)=P(f(X)\le y). When can we say that f(x)\le y\iff x\le f^{-1}(y) and therefore P(f(X)\le y)=P(X\le f^{-1}(y))? This holds, in particular, when f increases monotonically. Then f is injective, so f^{-1} exists.

    If X is non-negative, then P(X^2\le y)=P(X\le\sqrt{y}) because f(x)=x^2 increases monotonically when x\ge0. However, when X is normal, it can assume negative values. In this case, P(X^2\le y)=P(-\sqrt{y}\le X\le\sqrt{y})= F_X(\sqrt{y})-F_X(-\sqrt{y}). If X\sim\mathcal{N}(0,\sigma^2), this equals 2(F_X(\sqrt{y})-F(0)). In general, if X\sim\mathcal{N}(\mu,\sigma^2), you can use the fact that F_X(y)=\Phi\left(\frac{y-\mu}{\sigma}\right) where \Phi is the PDF of \mathcal{N}(0,1).
    Thanks from Rafael
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  5. #5
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    Re: Probability Density Function of a function

    Dear emakarov,

    thanks a lot, I would like to understand this in more ditails so could you pleas
    suggest me abook where I can read about this issue?
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  6. #6
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    Re: Probability Density Function of a function

    Sorry, I have not been studying or teaching probability lately, so I don't know good textbooks. But this is very basic stuff: just the definition of PDF and properties of normal distribution. I think that every textbook on probability should cover this.
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