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Math Help - distribution manipulation

  1. #1
    Member garymarkhov's Avatar
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    distribution manipulation

    I'm massively confused about what you can do to distributions formulas, and I haven't seen a nice collection of the rules anywhere.

    So when you have something like  u \sim N(0, \sigma^2) , I understand that you can say that  \frac{u}{\sigma^2} \sim N(0, 1) or that  u^2 \sim \chi^2 . But I don't know why these rules are true and how many of these rules exist. Can someone please point me to a useful resource? How can I manipulate these distribution expressions?
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    Quote Originally Posted by garymarkhov View Post
    I'm massively confused about what you can do to distributions formulas, and I haven't seen a nice collection of the rules anywhere.

    So when you have something like  u \sim N(0, \sigma^2) , I understand that you can say that  \frac{u}{\sigma^2} \sim N(0, 1) or that  u^2 \sim \chi^2 . But I don't know why these rules are true and how many of these rules exist. Can someone please point me to a useful resource? How can I manipulate these distribution expressions?
    This is a time consuming question to answer properly.

    Most books on mathematical statistics will have something to say about it.

    And a book I particularly like: The Algebra of Random Variables by M.D. Springer (most university libraries should have it).

    I also suggest doing a Google search to find things like http://www.cs.unm.edu/~williams/cs530/gst2.pdf

    And of course if you search the Advanced Probability and Statistics subforum at MHF you will find plenty of examples that use a variety of techniques.
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    Member garymarkhov's Avatar
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    Quote Originally Posted by mr fantastic View Post
    This is a time consuming question to answer properly.

    Most books on mathematical statistics will have something to say about it.

    And a book I particularly like: The Algebra of Random Variables by M.D. Springer (most university libraries should have it).

    I also suggest doing a Google search to find things like http://www.cs.unm.edu/~williams/cs530/gst2.pdf

    And of course if you search the Advanced Probability and Statistics subforum at MHF you will find plenty of examples that use a variety of techniques.
    Thanks for the book suggestion. I'll check it out.

    Any chance you could suggest three rules that are most commonly used and that I might find useful?
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    Using moment generating/characteristic functions, which characterize a distribution & using the Jacobian transformation
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    Quote Originally Posted by garymarkhov View Post
    Thanks for the book suggestion. I'll check it out.

    Any chance you could suggest three rules that are most commonly used and that I might find useful?
    1. Method of Distribution Functions.

    2. Method of Transformations.

    3. Method of Moment-Generating/Characteristic Functions.
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