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Thread: Super Tough Probability Question

  1. #1
    Junior Member
    Nov 2009

    Super Tough Probability Question

    Last edited by Porter1; Nov 29th 2009 at 08:35 PM.
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  2. #2
    MHF Contributor matheagle's Avatar
    Feb 2009
    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
    Super Member
    Mar 2008
    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.
    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
    $\displaystyle \Gamma(1/2) = \sqrt{2} \; \int_0^\infty \exp(-x^2/2) \, dx$
    which is equal to
    $\displaystyle \sqrt{2} \cdot \sqrt{2 \pi} \cdot \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$
    Based on your knowledge of the Normal distribution, you should know the value of $\displaystyle \frac{1}{\sqrt{2 \pi}} \; \int_0^{\infty} \exp(-x^2/2) \, dx$.

    Take it from there.
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