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Math Help - Properties of Expecation

  1. #1
    Junior Member
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    Properties of Expecation

    Hi,

    I have a problem in understanding the following property of expecation:

    see attached

    Why does E(X) only exist if this condition holds?

    Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

    Thanks
    Attached Thumbnails Attached Thumbnails Properties of Expecation-property-expectation.jpg  
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  2. #2
    Grand Panjandrum
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    Re: Properties of Expecation

    Quote Originally Posted by Marmy View Post
    Hi,

    I have a problem in understanding the following property of expecation:

    see attached

    Why does E(X) only exist if this condition holds?

    Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

    Thanks
    E(X)=\int_{-\infty}^{\infty} xf(x) dx

    which is an improper integral, and means:

    E(X)=\lim_{A,B \to \infty}\int_{-A}^{B} xf(x) dx=\lim_{A\to \infty}\int_{-A}^{0} xf(x) dx + \lim_{B \to \infty}\int_{0}^{B} xf(x) dx

    So the expectation exist iff the two limits on the right exist.

    Now try integration by parts.

    CB
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  3. #3
    Grand Panjandrum
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    Re: Properties of Expecation

    Quote Originally Posted by Marmy View Post

    Also, the formula always gives 0, no? Because the integral of F(X) from - infinity to + infinity should be 1.

    Thanks
    No, F(x) is the cumulative distribution, integral of F(X) from - \infty to + \infty should is infinite.

    F(x)=\int_{-\infty}^x f(\xi)\; d\xi

    CB
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  4. #4
    Junior Member
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    Re: Properties of Expecation

    Thanks! Great help!
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