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Positive integral for two functions

  1. May 13, 2015 #1

    ChrisVer

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    I am feeling stupid today...so:
    Is the following statement true?

    [itex] \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 [/itex]
    if [itex] f(x) \ge 0 [/itex] and [itex]0< \int_{-\infty}^{+\infty} g(x) < \infty [/itex] (integral converges)?

    If yes, then how could I show that? (It's not a homework) I am trying to understand how the expectation value [itex]E[XY] = <X,Y>[/itex] for two variables [itex]X,Y[/itex] can indeed be that inner product. Which would need that [itex]<X,X>=E[X^2] \ge 0 \Rightarrow \int_{-\infty}^{+\infty} x^2 g(x) dx \ge 0[/itex]. Here [itex]f(x)=x^2 >0[/itex] and [itex]\int g(x) = 1[/itex] (normalized pdf). Obviously it's true since all the negative values of the [itex]X[/itex] will become positive and so their expectation should be positive, but I'm trying to look in it mathematically.
     
    Last edited: May 13, 2015
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  3. May 13, 2015 #2

    PeroK

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    Suppose f(x) is 0 where g(x) is +ve. Can you get a counterexample from that?
     
  4. May 13, 2015 #3

    ChrisVer

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    If [itex]f(x)=0[/itex] then the equal sign holds...?
     
  5. May 13, 2015 #4

    PeroK

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    I don't think you understood my idea. f(x) = 0 where g(x) is +ve and f(x) > 0 where g(x) is -ve.

    It's not that hard to see.
     
  6. May 13, 2015 #5

    ChrisVer

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    I see... those statements are not sufficient then.... :/
     
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