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Math Help - Probability Questions

  1. #1
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    Post Probability Questions

    Hi,

    I have been trying to work this question for two days, but I could not figure it out. If any one help me to direct me that would be much appreciated. please!!!

    Suppose X follows a Normal distribution with mean (mu) (with mu>0) and a variance that is some function of the mean, (ie) (sigma^2) = h(u). Find h(u) such that P(X less than or equal to 0) does not depend on mean (mu).
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  2. #2
    Moo
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    Hello,

    Well, I'm not too sure about my method, but I get a result...

    \mathbb{P}(X\leq 0)=\int_{-\infty}^0 \frac{1}{\sqrt{2\pi} \cdot \sqrt{h(u)}} \cdot \exp\left(-\frac{(x-u)^2}{2h(u)}\right) ~dx

    Now substitute t=\frac{x-u}{\sqrt{h(u)}}

    You'll get :

    \mathbb{P}(X\leq 0)=\int_{-\infty}^{-u/\sqrt{h(u)}} \frac{1}{\sqrt{2\pi}} \cdot \exp\left(-\frac{t^2}{2}\right) ~dt

    So the only place left where there's u is in the boundary of the integral.
    Find any function h such that -\frac{u}{\sqrt{h(u)}}=\text{constant} (the only way not to have u)

    Spoiler:
    h(u)=au^2, for any positive constant a
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  3. #3
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    Many Thanks

    Many thanks for the help.

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