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Thread: Maybe a classical integral of normal distribution

  1. #1
    Mar 2012

    Maybe a classical integral of normal distribution

    I am a PhD student (not in maths). I have recently encountered a problem in my research which requires a solution to the following integral question:

    ================================================== =========================
    $\displaystyle f(x)=\frac{1}{\sigma _{1}\sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu _{1}}{\sigma _{1}}\, )^{2}}$

    $\displaystyle \phi (x)=\int_{-\infty }^{x} f(x)\, dx$

    $\displaystyle g(x)=\frac{1}{\sigma _{2}\sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu _{2}}{\sigma _{2}}\, )^{2}}$

    Find an expression or approximate expression of several terms for the following integral:

    $\displaystyle Pr(\mu _{1}\, , \sigma _{1}\, ,\mu _{2}\, , \sigma _{2}\)=\int_{-\infty}^{\infty} \phi (x) g(x) \, dx $

    ================================================== ==========================

    It looks like a classical question, and must have been solved by mathematicians long time ago. However I am not a maths student, so I don't know.
    If someone in this forum knows, please tell me. Exact solution or approximate solution with several terms are all OK.
    Any solution must be in general form involving those 4 parameters.
    Or please tell me about any journal article or maths textbook that discuss this question or similar question. Thank you!
    Last edited by wzhuang; Mar 13th 2012 at 05:38 PM.
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