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Math Help - help with derivation

  1. #1
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    [SOLVED] help with derivation

    I have spent hours on this and cant see where Im going wrong. First before I explain my problem here is the formula that I am working with...

    \tau^2=\frac{Q-n+1}{C}

    where

    Q=\sum^{n}_{i=1}(w_{i}\theta_{i}^2)-\frac{(\sum^{n}_{i=1}(w_{i}\theta_{i}))^2}{\sum^{n  }_{i=1}(w_{i})}

    C=\sum^{n}_{i=1}(w_{i})-\frac{\sum^{n}_{i=1}(w_{i}^2)}{\sum^{n}_{i=1}(w_{i  })}

    w_{i} and \theta_{i} are known values
    w_{i}=w_{1}, w_{2}, ... , w_{n}
    \theta_{i}=\theta_{1}, \theta_{2}, ... , \theta_{n}

    I want to know how to change \tau^2 to a set value by adding another \theta and w, which I will denote \theta_0 and w_0. I appreciate this is a 2-dimensional problem so I would like to know the value of \theta_0 which would make \tau^2= a given value if I also set w_0 to be a given value. To do this I have tried to derive an equation in the form \theta_0=f(\tau^2,w_0) where f is some function.

    Given the 'extra values' \theta_0 and w_0, the formula for \tau^2 now becomes...

    \tau^2=\frac{Q-n}{C}

    where

    Q=\sum^{n}_{i=1}(w_{i}\theta_{i}^2)+w_0\theta_0^2-\frac{(\sum^{n}_{i=1}(w_{i}\theta_{i})+w_0\theta_0  )^2}{\sum^{n}_{i=1}(w_{i})+w_0}

    C=\sum^{n}_{i=1}(w_{i})+w_0-\frac{\sum^{n}_{i=1}(w_{i}^2)+w_0^2}{\sum^{n}_{i=1  }(w_{i})+w_0}

    w_{i} and \theta_{i} are known values
    w_{i}=w_{1}, w_{2}, ... , w_{n}
    \theta_{i}=\theta_{1}, \theta_{2}, ... , \theta_{n}

    To simplify things, let

    v=\sum^{n}_{i=1}(w_{i}^2)
    x=\sum^{n}_{i=1}(w_{i}\theta_{i})
    y=\sum^{n}_{i=1}(w_{i})
    z=\sum^{n}_{i=1}(w_{i}\theta_{i}^2)

    which can all be evaluated given we have values for w_{i} and \theta_{i}. So, now for the derivation...

     \tau^2= \frac{z+w_0\theta_0^2-\frac{(x+w_0\theta_0)^2}{y+w_0}-n}{y+w_0-\frac{v + w_0^2}{y+w_0}}

     \tau^2(y+w_0-\frac{v + w_0^2}{y+w_0}) = z+w_0\theta_0^2-\frac{(x+w_0\theta_0)^2}{y+w_0}-n

     \tau^2(y+w_0-\frac{v + w_0^2}{y+w_0}) = \theta_0^2(\frac{yw_0}{y+w_0})-\theta_0(\frac{2xw_0}{y+w_0}) +z - n - \frac{x^2}{y+w_0}

     \tau^2(y+w_0-\frac{v + w_0^2}{y+w_0})(\frac{y+w_0}{yw_0}) = (\theta_0-\frac{x}{y})^2 + (\frac{y+w_0}{yw_0}) (z - n - \frac{x^2}{y+w_0})-0.25(\frac{2x}{y})^2

     \tau^2(y+w_0-\frac{v + w_0^2}{y+w_0})(\frac{y+w_0}{yw_0}) -(\frac{y+w_0}{yw_0}) (z - n - \frac{x^2}{y+w_0})+0.25(\frac{2x}{y})^2= (\theta_0-\frac{x}{y})^2

     \theta_0=\frac{x}{y} \pm \sqrt(\tau^2(y+w_0-\frac{v + w_0^2}{y+w_0})(\frac{y+w_0}{yw_0}) -(\frac{y+w_0}{yw_0}) (z - n - \frac{x^2}{y+w_0})+(\frac{x}{y})^2)

    Everything seems right but when I test it by finding \theta_0 and substituting back into the \tau^2 formula I am not getting the right result for \tau^2. Where am I going wrong???

    Any help much appreciated.
    Last edited by deanj2k; February 5th 2010 at 01:18 AM. Reason: Solved
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