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Math Help - Rayleigh pdf ratio and parameter estimation

  1. #1
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    Rayleigh pdf ratio and parameter estimation

    Dear all,
    I've a problem regarding Rayleigh pdf.
    In particular, I know the ratio X between two part of the probability density function, at the right and at the left of a given threshold point th, so

    \int_{th}^{\infty}f(x,\sigma)dx=X\int_{0}^{th}f(x,  \sigma)dx

    and from the former I need to compute the value of the parameter \sigma of the Rayleigh distribution f(x,\sigma).

    I've made the first steps, reducing the former in

    \int_{0}^{th}f(x,\sigma)dx=\frac{1}{1+X}

    and solving the definite integral, obtaining

    \frac{\frac{1}{\sigma^{2}}\left(  e^{\frac{-th^{2}}{2\sigma^{2}}}-1\right)<br />
}{1-\frac{1}{\sigma^{2}}}=\frac{1}{1+X}

    However, is now possible to solve the last equation in \sigma? otherwise, is there any other method to compute (or, at least, estimate) the value of \sigma from the first equation?

    Thank you in advance for any suggestion!
    Simo
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  2. #2
    Moo
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    Hello,

    The cumulative density function of a Rayleigh distribution is (according to the wikipedia) :

    \int_0^t f(x,\sigma) ~dx=1-\exp\left(\frac{-t^2}{2\sigma^2}\right)

    So you actually have :

    1-\exp\left(\frac{-th^2}{2\sigma^2}\right)=\frac{1}{1+X}

    But it looks like you don't have the same definition of the Rayleigh distribution than the wikipedia...
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  3. #3
    MHF Contributor matheagle's Avatar
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    That's why I wanted more info.
    A lot of these densities can be written several ways.
    These posters need to write down their densities instead of having me guess how their parameters are written.
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  4. #4
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    Moo,
    you are absolutely right!
    ...and this solution is also more "tractable"

    Thank you very much!
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