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Math Help - Distribution Theory

  1. #1
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    Distribution Theory

    If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2
    >>Jacques
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  2. #2
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    Quote Originally Posted by JacquesRoux View Post
    If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2
    >>Jacques
    I don't have the time right now to post a solution. But if you read number 10 of this you just might figure it out yourself ....

    If you're still stuck, please say where - if no-one else replies I'll post some help in the morning (my morning!)
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  3. #3
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    Thanks very much will let you know if I need more help!
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  4. #4
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    Distribution Theory

    C = Integral R exp(-z2 / 2)dz = (2 )1/2.
    Hint: Express C^2 as a double integral over R^2 and then convert to polar coordinates:

    Is this the correct method for the above problem. If it is the correct one how will I determine the intervals of the double integral. I am really stuck with this problem and really will appreciate guidedance with this problem. Please help me!!
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  5. #5
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    Quote Originally Posted by JacquesRoux View Post
    C = Integral R exp(-z2 / 2)dz = (2 )1/2.
    Hint: Express C^2 as a double integral over R^2 and then convert to polar coordinates:

    Is this the correct method for the above problem. If it is the correct one how will I determine the intervals of the double integral. I am really stuck with this problem and really will appreciate guidedance with this problem. Please help me!!
    I'm sorry but

    C = Integral R exp(-z2 / 2)dz = (2 )1/2.
    Hint: Express C^2 as a double integral over R^2 and then convert to polar coordinates:

    is unclear to me. What is R? Does exp(-z2 / 2) mean e^{-z^2/2}\, ? What does (2 )1/2 mean? It would help if you could typeset these sorts of complex expressions using latex. Where has the hint come from?
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    Red face Distribution Theory

    Sorry this was the question:

    If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2

    and I have no clue how to even start the problem. Can you please give me some guidedance on how to even begin solving this problem.
    Jacques
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  7. #7
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    Quote Originally Posted by JacquesRoux View Post
    Sorry this was the question:

    If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2

    and I have no clue how to even start the problem. Can you please give me some guidedance on how to even begin solving this problem.
    Jacques
    Let F(x) = \Pr(\chi^2 < x)


    \Rightarrow F(x) = \Pr(-\sqrt{x} < \chi < \sqrt{x})


    = \frac{1}{\sqrt{2 \pi}} \, \int_{-\sqrt{x}}^{\sqrt{x}} e^{-u^2/2} \, du


    = \frac{2}{\sqrt{2 \pi}} \, \int_{0}^{\sqrt{x}} e^{-u^2/2} \, du.


    Therefore f(x) = \frac{dF}{dx} = \frac{2}{\sqrt{2 \pi}} \, e^{-x/2} \, \left (\frac{1}{2 \sqrt{x}} \right) \, , x > 0

    where the Fundamental Theorem of Calculus and the chain rule have been used to get the derivative


    = \frac{1}{\sqrt{2 \pi}} \, e^{-x/2} \, \frac{1}{\sqrt{x}} \, , x > 0


    and f(x) = 0 for x < 0.

    As expected, this is the pdf for the chi-square distribution of degree 1. Note: \frac{1}{\sqrt{2 \pi}} = \frac{1}{\Gamma(1/2) 2^{1/2}}.


    You could also find the moment generating function of \chi^2 and recognise it as being the moment generating function of the chi-square distibution of degree 1.
    Last edited by mr fantastic; March 23rd 2008 at 05:34 PM. Reason: Added the moment generating function bit
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  8. #8
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    Distribution Theory

    Thanks very much!!

    Jacques
    Last edited by mr fantastic; January 15th 2009 at 03:53 AM. Reason: Removed reference to first name
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