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Math Help - Conditional expectation with inequality

  1. #1
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    Conditional expectation with inequality

    Hi, so the definition of E(g(Y_1)|Y_2=y_2) = \int_{-\infty}^{\infty} g(y_1) f(y_1|y_2) dy_1 however how do we define E(g(Y_1)|Y_2 \le y_2) ie, when the condition is an inequality rather than an equality?

    Thanks
    Last edited by usagi_killer; February 13th 2013 at 08:51 PM.
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  2. #2
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    Re: Conditional expectation with inequality

    Hey usagi_killer.

    For the inequality condition what is done is we find a random variable that fits the definition of Y2 <= y2.

    The easiest way to do this is have a double integral with the limits of Y2 going from -infinity to y2.

    When you evaluate this double integral you will get an expression in terms of y2.
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  3. #3
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    Re: Conditional expectation with inequality

    Thanks chiro, so what would the expression look like? Do we still use the conditional density function?

    Would it be:

    E(g(Y_1)|Y_2 \le y_2) =   \int_{-\infty}^{y_2} \int_{-\infty}^{\infty} g(y_1) f(y_1|y_2) dy_1 dy_2

    thanks
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  4. #4
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    Re: Conditional expectation with inequality

    Yes that looks right.
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