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Math Help - Likelihood function derivation help needed

  1. #1
    da`
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    Likelihood function derivation help needed

    struggling with this. can anyone point me in the right direction?

    see attached, i couldnt write down the function at the bottom in here to made it and had to attach it.
    Last edited by da`; November 27th 2008 at 01:43 PM.
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  2. #2
    Flow Master
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    Quote Originally Posted by da` View Post
    struggling with this. can anyone point me in the right direction?

    see attached, i couldnt write down the function at the bottom in here to made it and had to attach it.
    L \propto \left[ exp\left(-\frac{x_1^2}{2}\right) \cdot exp\left(-\frac{x_2^2}{2}\right) \cdot \, .... \cdot exp\left(-\frac{x_{\theta}^2}{2}\right) \right] \cdot \left[exp\left(-\frac{(x_{\theta + 1}^2 - 1)}{2}\right) \cdot \, .... \cdot exp\left(-\frac{(x_{n}^2 - 1)}{2}\right) \right]


    = exp \left(- \sum_{i=1}^{\theta} \frac{x_i^2}{2} \right) \cdot exp \left(- \sum_{i=\theta + 1}^{n} \frac{(x_i^2 - 1)}{2} \right)


    = exp \left(-\sum_{i=1}^{\theta} \left[\frac{(x_i - 1)^2}{2} + \frac{(2x_i - 1)}{2}\right] \right) \cdot exp \left(- \sum_{i=\theta + 1}^{n} \frac{(x_i^2 - 1)}{2} \right)


    = exp \left(- \sum_{i=1}^{n} \frac{(x_i - 1)^2}{2}\right) \cdot exp \left(- \sum_{i=1}^{\theta} \frac{(2x_i - 1)}{2} \right)


    \propto exp \left(- \sum_{i=1}^{\theta} \frac{(2x_i - 1)}{2} \right).
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  3. #3
    da`
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    Thank you very much, i understand.
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