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Math Help - Super Tough Probability Question

  1. #1
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    Super Tough Probability Question

    ........................
    Last edited by Porter1; November 29th 2009 at 08:35 PM.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Just do (a) directly with polar co-ordinates.
    That's how you prove the normal density integrates to one.
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  3. #3
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    Quote Originally Posted by Porter1 View Post
    Hi all, I have this question and I really need your help!

    (a) Show that Г(1/2) = √(П) (<-Square root of Pi) by writing
    Г(1/2) = ∫(0)->(∞) y^(-1/2) e^(-y) dy
    by making the transformation y = (x^2)/2 and using the standard normal density.
    [snip]
    You can work this out from basics as suggested by MathEagle, but if you are allowed to assume some basic knowledge of the Normal distribution there is a shortcut, which I think is probably what you are intended to do--

    After you make the substitution, you should have
    \Gamma(1/2) = \sqrt{2} \; \int_0^\infty \exp(-x^2/2) \, dx
    which is equal to
    \sqrt{2} \cdot \sqrt{2 \pi} \cdot \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx
    Based on your knowledge of the Normal distribution, you should know the value of \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx.

    Take it from there.
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