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Math Help - Problem on Random Variables

  1. #1
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    Problem on Random Variables

    I'm working on the following problem for a probability class.

    Let X_1, X_2,... be independent random variables with a common continuous distribution function. Let B be the \omega-set where X_m(\omega)=X_n(\omega) for some pair m,n of distinct integers, and show that P(B)=0.

    Part of my problem is that this section we're studying on random variables introduces so many new terms that I haven't been able to see how they work together very well. I understand that P(B)=0 means that X_m(\omega)\ne X_n(\omega) for any m\ne n except on a set of probability (measure) 0, but am struggling mostly with the first sentence. I'm not asking for a full fledged solution to the problem, but I would be grateful for any assistance on understanding the interplay between the other terms involved and advice on how to start.
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  2. #2
    Moo
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    Hello,

    You can try to find the pdf of X_m-X_n (by first finding the pdf of (Y,Z)=(X_m-X_n,X_n) thanks to a Jacobian transformation and then integrating)

    Then, you'll get something like g(y)=\int f(y+z)f(z) ~dz for the pdf of Y.

    P(Y=0)=\int_{\{0\}} \int f(y+z)f(z) ~dz ~dy

    Reverse the integration order (what's in the integral is positive so we can apply Fubini-Tonelli's theorem) and you'll get 0, because you integrate over a set of Lebesgue measure 0.


    Otherwise, maybe you can have a look into the properties of convolution... But right now I have to go to school... Good luck !
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