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Math Help - probaility analysis

  1. #1
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    probaility analysis

    let f be a continuous density which is zero outside in the interval[0,A]
    ,where A >0, let F be the corresponding cdf.

    a) prove that there exists at least one point X such that f(x)=1/A
    b), let 0<p<1 prove there exists at least one point Xp such that Xp is the pth quantile of the distribution defined by f and F, give an example to show that the point Xp may not be unique.


    Thanks a lot!!
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  2. #2
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    Hint for part a. Do you know the mean value theorem for integrals?
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  3. #3
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    Quote Originally Posted by minmin View Post
    let f be a continuous density which is zero outside in the interval[0,A]
    ,where A >0, let F be the corresponding cdf.

    a) prove that there exists at least one point X such that f(x)=1/A
    You can't prove it- it's not true. For example, if A= 1/2, F(x) is always between 0 and 2 while 1/A= 2.

    b), let 0<p<1 prove there exists at least one point Xp such that Xp is the pth quantile of the distribution defined by f and F, give an example to show that the point Xp may not be unique.


    Thanks a lot!!
    What do you mean by "quantile"? A point where the probability P(x< Xp)= p? In that case you just need the fact that F is a continuous function, F(0)= 0, F(A)= 1 and the "intermediate value property" of continuous functions.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    You can't prove it- it's not true. For example, if A= 1/2, F(x) is always between 0 and 2 while 1/A= 2.


    What do you mean by "quantile"? A point where the probability P(x< Xp)= p? In that case you just need the fact that F is a continuous function, F(0)= 0, F(A)= 1 and the "intermediate value property" of continuous functions.

    Hi the quantile means the inverse of cdf. i think the point Xp is the unique. how can i prove it?
    thanks
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