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Math Help - Prove that it's real

  1. #1
    Moo
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    Prove that it's real

    Hi,

    I've had this in an exam, and I must say I can't see how to prove it...

    F(u)=\int_{\mathbb{R}} \exp\left(-\tfrac{x^2}{2}+iux\right) ~dx

    How to prove that it's a real-valued function ?

    ( u\in\mathbb{R})


    Thanks
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    Quote Originally Posted by Moo View Post
    Hi,

    I've had this in an exam, and I must say I can't see how to prove it...

    F(u)=\int_{\mathbb{R}} \exp\left(-\tfrac{x^2}{2}+iux\right) ~dx

    How to prove that it's a real-valued function ?

    ( u\in\mathbb{R})


    Thanks
    It's real if it equals its conjugate... In this case, symmetry x\mapsto -x shows that it is the case.

    In general, if a distribution is symmetric (X has same distribution as -X), then its characteristic function is real. This is even an equivalent condition since we always have (for a real valued r.v.) \overline{\Phi_{X}(t)}=\Phi_{-X}(t) for all t.
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    Moo
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    Quote Originally Posted by Laurent View Post
    It's real if it equals its conjugate... In this case, symmetry x\mapsto -x shows that it is the case.

    In general, if a distribution is symmetric (X has same distribution as -X), then its characteristic function is real. This is even an equivalent condition since we always have (for a real valued r.v.) \overline{\Phi_{X}(t)}=\Phi_{-X}(t) for all t.

    That was easy...

    Yeah, I know the part for the characteristic function, but it wasn't a probability exam (though I quickly tried to find a relationship with it, en vain), so I wasn't programmed for thinking about this
    It would even have helped me for another question in the same exercise...



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