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Math Help - Proving the second property of Density Functions

  1. #1
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    Proving the second property of Density Functions

    hi


    I'm trying to prove the four properties of the density functions.

    The question goes, "Show that the properties of a density function f_X(x) are valid."

    I have done the first one, but I am having trouble with the second one.
    The second property of density functions in my books goes like this....

    \int^\infty_{-\infty}f_X(x)dx = 1

    can anyone help?
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  2. #2
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    Quote Originally Posted by Bucephalus View Post
    hi


    I'm trying to prove the four properties of the density functions.

    The question goes, "Show that the properties of a density function f_X(x) are valid."

    I have done the first one, but I am having trouble with the second one.
    The second property of density functions in my books goes like this....

    \int^\infty_{-\infty}f_X(x)dx = 1

    can anyone help?
    Remember that a probability must satisfy P(\Omega)=1 where \Omega is the whole probability space... In your case, \Omega=\mathbb{R}.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Do you have a specific density?
    Or are you asking why a density must integrate to one?
    Since probability=area, the total area=total probability=1.
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  4. #4
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    Where to from here

    Thanks,

    How do I say this though.
    Do I just say that f_X(x) represents the probability space which is 1. Therefore the area under the curve is 1.

    I was trying to prove in terms of the integral result which is also  F_X(x). A function that extends from -\infty at a value of 0 to \infty at a value of 1.
    Is there anything in this function that would suggest that it's derivative.
    I think it just dropped...does this sound like a valid.
    the function that is the integral of f_X(x) is F_X(x) and if we take the values of F_X(x) at \infty and -\infty, i.e. the limits, it will evaluate to 1 - 0 = 1.

    Is that a valid validation?
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  5. #5
    Moo
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    Hello,

    \mathbb{P}(X\in A)=\int_A f_X(x) ~dx by definition.

    Hence \int_{-\infty}^\infty f_X(x) ~dx=\mathbb{P}(X\in\mathbb{R})

    But (you didn't state it but we can assume) X has its values in \mathbb{R}. So, as Laurent said, \Omega=\mathbb{R}

    I think that since you're asked to prove the properties of the pdf, you wouldn't be asked to use the cdf...
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  6. #6
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    Thanks to all

    Thanks for you help.
    It's starting to make sense now.
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