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Math Help - Using the weak law of large numbers

  1. #1
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    Using the weak law of large numbers

    Hello,
    To give a brief background
    in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

    This is one of the examples given in the book, but I just can't seem to understand it. I especially do not understand the boxed step, and why the expected value of f(Ui) equals that integral.

    If someone could provide a detiailed explanation, I would be very grateful.
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    Last edited by CaptainBlack; November 25th 2007 at 12:11 AM.
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  2. #2
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    Quote Originally Posted by johnj77 View Post
    Hello,
    To give a brief background
    in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

    This is one of the examples given in the book, but I just can't seem to understand it. I especially do not understand the boxed step, and why the expected value of f(Ui) equals that integral.

    If someone could provide a detiailed explanation, I would be very grateful.
    By definition of f and U_i :

    E(|X_i|) = E(|f(U_i)|)

    E(|f(U_i)|) = \int_0^1 |f(x)| dx

    as again by given condition U_i \sim U(0,1).

    But by definition f(x) \ge 0 so we have:

    E(|X_i|) = E(|f(U_i)|)= \int_0^1 |f(x)| dx= \int_0^1 f(x) dx<\infty

    the last inequality because again by definition f is integrable

    RonL
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