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Math Help - Simple if not fot the Trick!

  1. #1
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    Simple if not fot the Trick!

    If it is already given that X1 and X2 are independent R.V. each having p.d.

    f(x) = { e^-x , x > 0
    { 0 , otherwise

    How would one go about showing that R.V. (s) R.V.s Y1 and Y2 are independent when Y1 = X1 + X2, Y2 = \frac{X1}{X1 + X2}

    Do I need the abstraction of Y1 and Y2 or can I just leave it in terms of X? I'm not sure if thats the right question. The additional R.V.s are throwing me off.

    Oh, and can someone give me a link to a tutorial or listing of the Math Markup used to display Math notation so my next thread or answer can look better.

    Thank you
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  2. #2
    MHF Contributor matheagle's Avatar
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    I don't know what this means...Do I need the abstraction of Y1 and Y2 or can I just leave it in terms of X?

    You need to make a two dimensional change.
    It's just calc 3.
    The joint density of X_1 and X_2 is the product of their marginals.
    So, f(x_1,x_2)=exp{-(x_1+x_2)} in the irst quadrant, x_1>0, x_2>0.
    NOW ge the jacobian, which you should recall is the determinant of the four partial derivative.
    INSERT your new rvs, y_1 and y_2 and multiple by the absolute value of your jacobian and you're done.
    Clearly, since X_1 and X_2 are exponentials, which are gamma (1,1)
    We have via MGFs that the sum is a gamma (2,1)
    And one can see that Y_2 is between 0 and 1, hence it's Beta with mean .5.
    I think it's a Beta(1,1).
    But to see that they are independent you need the joint density of Y_1 and Y_2.


    {x\over x+y}, just learning your TeX here
    Last edited by matheagle; February 21st 2009 at 10:54 PM.
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  3. #3
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    Quote Originally Posted by Haskell View Post
    ...
    Oh, and can someone give me a link to a tutorial or listing of the Math Markup used to display Math notation so my next thread or answer can look better.

    Thank you
    See the tutorial here:

    http://www.mathhelpforum.com/math-he...-tutorial.html

    Math Help has an entire forum devoted to LaTex:

    http://www.mathhelpforum.com/math-help/latex-help/

    Also, you can see other user's LaTex by simply pressing the Quote button at the end of a post.
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